Matrix Methods in Analysis

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1113 II I

III I

Piotr Antosik Charles Swartz

Matrix Methods in Analysis

Springer-Verlag Berlin Heidelberg New York Tokyo 1985

Authors

Piotr Antosik Institute of Mathematics, Polish Academy of Sciences ul. Wieczorka 8, 40-013 Katowice, Poland Charles Swartz Department of Mathematical Sciences, New Mexico State University Las Cruces, N M . 88003, USA

A M S Subject Classification (1980): Primar~ 46-0, Secondary. 2 8 C 20, 4 0 A 0 5 , 4 0 C 0 5 , 46A15, 4 6 G 1 0 ISBN 3-540-15185-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15185-0 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. Atl rights are reserved,whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "VerwertungsgesellschaftWort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-5432t0

Contents

1.

Introduction

. . . . . . . . . . . . . . . . . . . . . . . .

Introductory

remarks, outline,

notation

and t e r m i n o l o g y ,

Drewnowski Lemma.

2.

Basic Matrix Results

. . . . . . . . . . . . . . . . . . . .

7

. . . . . . . . . . . . . . . . . . . . . .

11

B a s i c M a t r i x Theorem.

.

X -convergence Definition

of

K-convergence,

K-boundedness,

examples,

K-convergence doesn't

K - c o n v e r g e n c e and

K-space,

imply

K-bounded,

K - b o u n d e d n e s s in t h e weak and weak~

topologies.

4.

The U n i f o r m B o u n d e d n e s s P r i n c i p l e

. . . . . . . . . . . . .

General Uniform Boundedness Principle, Boundedness Principle,

.

. . . . . . . . . . . . . . . . . .

Classical

Banach-Steinhaus

Steinhaus

Theorem, t h e B r o o k s - J e w e t t

Bilinear

continuity,

separate

separate

T h e o r e m s , t h e Hahn-

. . . . . . . . . . . . . .

continuity

continuity

for sequences of bilinear tinuity

Theorem, Nikodym

Theorem.

Maps and H y p o c o n t i n u i t y

K-hypocontinuity,

29

Theorems, General Banach-

C o n v e r g e n c e and V i t a l i - H a h n - S a k s

.

Uniform

Nikodym B o u n d e d n e s s Theorem.

Convergence of Operators

Schur Su~ability

Classical

21

implies

and e q u i c o n t i n u i t y

maps, s e p a r a t e

and e q u i h y p o c o n t i n u i t y .

joint

equicon-

45

Iv 7.

Orlicz-Pettis Classical Pettis

Theorems . . . . . . . . . . . . . . . . . . .

Orlicz-Pettis

Theorem, T w e d d l e - t y p e O r l i c z -

Theorem, O r l i c z - P e t t i s

Orlicz-Pettis

58

result

f o r compact o p e r a t o r s

Theorems f o r t h e t o p o l o g y o f p o i n t w i s e

c o n v e r g e n c e in f u n c t i o n

spaces,

abstract

Orlicz-Pettis

Theorems.

8.

The S c h u r and P h i l l i p s

. . . . . . . . . . . . . . .

G e n e r a l S c h u r Lemma, C l a s s i c a l

S c h u r Lemma, G e n e r a l

Hahn-Schur Summability R e s u l t ,

General Phillips'

Lemma, C l a s s i c a l

9.

Le~mas

Phillips"

74

Lemma.

The Schur Lemma for Bounded Multiplier Convergent Series .

85

General Schur Lemma for Bounded Multiplier Convergent Series, Summability Result, C-convergent series, Schur Lemma for C-convergent series.

I0.

Imbedding c o and ~

. . . . . . . . . . . . . . . . . . . .

Bessaga-Pelczynski

Weak u n c o n d i t i o n a l

Cauchy s e r i e s ,

Theorem, D i e s t e l ' s

Theorems, P e l c z y n s k i ' s

unconditionally

94

converging operators,

Theorem on

Diestel-Faires

Theorem, B e s s a g a - P e l c z y n s k i Theorem, R o s e n t h a l ' s Theorem.

References

Index

Notation

. . . . . . . . . . . . . . . . . . . . . . . . . . .

105

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

110

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 113 . .

.

1.

Introduction

In t h i s results

set of tecture

on i n f i n i t e

notes,

we p r e s e n t

m a t r i c e s w h i c h were e v o l v e d by t h e members o f t h e

Katowice Branch of the Mathematics Institute Sciences.

In t h e e a r l y

history

methods were used e x t e n s i v e l y results

a culmination of

in f u n c t i o n a l

o f t h e P o l i s h Academy o f

of functional

to establish

analysis.

sliding

"sliding

some o f t h e e a r l y

For example,

v e r s i o n s o f t h e U n i f o r m Boundedness P r i n c i p l e Hildebrand utilized

analysis

the first

hump"

abstract

proofs of

by Hahn and Banach and

hump methods ( [ 1 8 ] ,

[39],

[42],

[35]).

S i n c e Banach and S t e i n h a u s g a v e a p r o o f o f t h e U n i f o r m B o u n d e d n e s s Principle

b a s e d on t h e B a i r e C a t e g o r y Theorem, c a t e g o r y methods have

p r o v e n t o be v e r y p o p u l a r i n t r e a t i n g analysis

[191.

In r e c e n t t i m e s ,

hump" methods in t r e a t i n g

F o r example,

Rosenthal

as an a b s t r a c t

variety

of topics

sliding

in f u n c t i o n a l

in [34] D i e s t e l sliding

[2],

[3],

in f u n c t i o n a l

fashion,

quite effective

hump method t o t r e a t

a

the Antosik-Mikusinski Diagonal

analysis

[5],

m a t r i c e s and has p r o v e n t o be

various topics

t h a t were p r e v i o u s l y

by B a i r e c a t e g o r y methods ( s e e in p a r t i c u l a r

I56]).

These n o t e s p r e s e n t a r e s u l t

the texts

concerning infinite

in f u n c t i o n a l

analysis

be used t o t r e a t

[12],

matrices

w h i c h i s o f an even s i m p l e r and more e l e m e n t a r y c h a r a c t e r D i a g o n a l Theorem, and w h i c h c a n s t i l l

[6],

The A n t o s i k - M i k u s i n s k i D i a g o n a l

concerning infinite

in t r e a t i n g

a wide v a r i e t y

and m e a s u r e t h e o r y ( [ 4 ] ,

treated

of topics

and

[ 9 ] ) can be c o n s i d e r e d t o be an a b s t r a c t

[9], [ 1 2 ] , [ 5 3 ] , [ $ 4 ] , [S61, [ 5 7 ] ) . Theorem i s a r e s u l t

analysis

and Uhl use a lemma o f

hump method and h a s been employed t o t r e a t

of topics

to "sliding

i n v e c t o r measure t h e o r y .

In a somewhat s i m i l a r Theorem ( [ 5 3 ] ,

in f u n c t i o n a l

t h e r e has been a r e t u r n

various topics

measure t h e o r y . ([64])

various topics

than the

a wide v a r i e t y

and m e a s u r e t h e o r y ( [ 1 6 ] ) .

In s e c t i o n

2, we p r e s e n t t h e two b a s i c m a t r i x r e s u l t s

P. A n t o s i k i n r e f e r e n c e s

[6] - [ I 1 1 , and t h e n in s u b s e q u e n t s e c t i o n s

we p r e s e n t v a r i o u s a p p l i c a t i o n s functional

analysis

been p r e s e n t e d

Thus,

there

of the matrix results

and measure t h e o r y .

in s e c t i o n s

the subsequent chapters

e v o l v e d by

to topics

in

After the basic material

has

2 and 3, t h e r e has been an a t t e m p t t o make

on a p p l i c a t i o n s

i s some r e p e t i t i o n

i n d e p e n d e n t o f one a n o t h e r .

in some o f t h e c h a p t e r s ;

s u m m a b i l i t y i s m e n t i o n e d in b o t h s e c t i o n s

f o r example,

S and 8 and o t h e r t o p i c s

are repeated. In s e c t i o n g e n c e and

K

3, we i n t r o d u c e and s t u d y t h e n o t i o n s o f

- [11]).

An e q u i v a l e n t

c o n v e r g e n c e was i n t r o d u c e d by S. Mazur and W. O r l i c z s t u d i e d by A. A l e x i e w i c z in [ 1 ] .

tions

K

c o n v e r g e n c e and

substitutes

example,

in s e c t i o n

The c l a s s i c a l

~

boundedness w i l l

results

4, we t r e a t

in t h e

In s u b s e q u e n t s e c t i o n s

assump-

analysis.

For

t h e U n i f o r n Boundedness P r i n c i p l e .

Uniform Boundedness P r i n c i p l e

i s w e l l - k n o w n t o be f a l s e assumptions,

a v e r s i o n o f t h e Uniform Boundedness P r i n c i p l e

but we

in Theorem 4 . 2

which i s v a l i d

in t h e a b s e n c e o f any c o m p l e t e n e s s a s s u m p t i o n and

which c o n t a i n s

the classical

as a s p e c i a l

case.

derivation

U n i f o r m Boundedness Theorem f o r F - s p a c e s

To i l l u s t r a t e

Boundedness P r i n c i p l e

the

be shown t o be

of functional

in t h e a b s e n c e o f completeness o r b a r r e l l e d n e s s present

in [52] and a l s o

f o r c o m p l e t e n e s s and b a r r e l l e d n e s s

in many o f t h e c l a s s i c a l

form o f

The i d e a was r e d i s c o v e r e d

s e m i n a r o f P. A n t o s i k and J . M i k u s i n s k i .

effective

conver-

b o u n d e d n e s s w h i c h were a l s o d i s c o v e r e d and s t u d i e d by

the Katowice mathematicians ([6]

notions of

~

the utility

of our general Uniform

in t h e a b s e n c e o f c o m p l e t e n e s s , we g i v e a

o f t h e Nikodym Boundedness Theorem b a s e d on t h e g e n e r a l

Uniform B o u n d e d n e s s P r i n c i p l e . In s e c t i o n

5, we d i s c u s s

a classical

result

on t h e c o n v e r g e n c e

of operators

w h i c h i s sometimes a t t r i b u t e d

t o Banach and S t e i n h a u s .

This result,

l i k e t h e Uniform Boundedness P r i n c i p l e ,

i s known t o be

false without completeness or barrelledness assumptions. less, using the notion of

K

Neverthe-

convergence, we present a version of

this theorem which is valid without any completeness assumptions.

As

an application of the general result in the absence of completeness, we use it to derive the Nikodym Convergence Theorem, the BrooksJewett Theorem, and a result of Hahn, Schur and Toeplitz on summability. In section 6, we treat bilinear maps using our matrix methods. We derive the classical result of Mazur and Orlicz on the joint continuity of separately continuous bilinear maps and also, using the notion of

~

convergence,

present several hypocontinuity type of

results which are valid without completeness assumptions.

Our

hypocontinuity results generalize results of Bourbaki. In section 7, we treat various 0rlicz-Pettis type results on subseries convergent series by matrix methods.

We derive the

classical Orlicz-Pettis Theorem as well as Orlicz-Pettis results for compact operators and the topology of pointwise convergence on certain well-known function spaces. In section 8, we give generalizations of the classical lemmas of Schur and Phillips to the group-valued case.

We show that these gen-

eral results contain the classical lemmas of Schur and Phillips as special cases.

A result of Hahn and Schur on summability is also

obtained from the general results. In section 9, we present a version of the Schur lemma for bounded multiplier convergent series in a metric linear space.

This

version for bounded multiplier convergent series is motivated by a sharper conclusion of the classical Schur lemma for is obtained in Corollary 8.4.

B-spaces which

Some general remarks on the vector

versions of the summability results of Schur and Hahn are also included.

In s e c t i o n into a

I 0 , we c o n s i d e r t h e p r o b l e m o f imbedding

B-space.

U s i n g t h e b a s i c m a t r i x lemma o f s e c t i o n

obtain the classical on imbedding cations

co

results

and

£~

The r e s u l t s

of Diestel

into

B-spaces.

and Uhl ( [ 3 4 ]

1.4) except t h a t

the matrix results

e l e m e n t a r y in c h a r a c t e r , topics

We a l s o i n d i c a t e

in Banach s p a c e

treated R

throughout these notes.

are extremely effective analysis

in t r e a t i n g

various

w h i c h have been

by B a i r e c a t e g o r y m e t h o d s .

The o t h e r theme i s

the idea of

c o n v e r g e n c e can be used as an e f f e c t i v e

tute

f o r c o m p l e t e n e s s a s s u m p t i o n s in many c l a s s i c a l analysis.

The

although being very

that

functional

appli-

t h e b a s i c m a t r i x lemma i s

presented here,

in measure t h e o r y and f u n c t i o n a l

traditionally

2, we

lemma.

T h e r e a r e two themes w h i c h p r e v a i l is that

~

and method o f p r o o f a r e v e r y a n a l o g o u s t o t h o s e

employed i n s t e a d o f t h e R o s e n t h a l

first

and

o f B e s s a g a - P e l c z y n s k i and D i e s t e l - F a i r e s

t o a l a r g e number o f w e l l - k n o w n r e s u l t s

theory.

co

substi-

results

For example, we p r e s e n t v e r s i o n s

in

of the Uniform

Boundedness P r i n c i p l e ,

t h e B a n a c h - S t e i n h a u s Theorem and c l a s s i c a l

hypocontinuity

w h i c h a r e v a l i d w i t h no c o m p l e t e n e s s assump-

results

tions whatever being present.

Applications

of these general

results

in t h e a b s e n c e o f c o m p l e t e n e s s a r e i n d i c a t e d . Many o f t h e t o p i c s functional tional

analysis

analysis

in t h e s e notes are s t a n d a r d t o p i c s

which are t r e a t e d

texts

c a t e g o r y methods.

treated

in a g r e a t number o f t h e f u n c -

by v a r i o u s means i n c l u d i n g t h e p o p u l a r B a i r e

The m a t r i x methods employed in t h e s e n o t e s a r e o f

a very elementary character

and can be p r e s e n t e d w i t h o u t r e q u i r i n g

g r e a t d e a l o f m a t h e m a t i c a l b a c k g r o u n d on t h e p a r t o f t h e r e a d e r . this

in

a For

r e a s o n t h e s e m a t r i x methods would seem t o be q u i t e a p p r o p r i a t e

for presentation

o f some o f t h e c l a s s i c a l

functional

t o r e a d e r s w i t h modest m a t h e m a t i c a l b a c k g r o u n d s . hope t h a t the future

t h e m a t r i x methods p r e s e n t e d h e r e w i l l functional

analysis

texts.

It

analysis

topics

is the authors"

find their

way i n t o

We c o n c l u d e t h i s

introduction

by f i x i n g

the n o t a t i o n which w i l l

be used i n t h e s e q u e l . Throughout the remainder of these notes, stated E

otherwise,

E, F

and

G

w i l l d e n o t e normed g r o u p s .

i s assumed t o be an A b e l i a n t o p o l o g i c a l

g e n e r a t e d by a q u a s i - n o r m ]01 = 0,

l-x]

= Ix[

]I

and

metric topology on

E

: E ÷ R+.

Ix+y] (

unless explicitly That i s ,

g r o u p whose t o p o l o g y i s ([I

is a quasi-norm if

Ix[ + ]Yl;

a quasi-norm generates

a

via the translation invariant metric

d(x,y) = lx-yl.) Recall that the topology of any topological group is always generated by a family of quasi-norms ([27]).

Thus, many of the results

are actually valid for arbitrary topological groups.

We present the

results for normed groups only for the sake of simplicity of exposition. Similarly,

X, Y

and

Z

will denote metric linear spaces whose

ll-

topologies are generated by a quasi-norm

(For convenience, all

vector spaces will be assumed to be real; most of the results are valid for complex vector spaces with obvious modifications.) is further assumed that the norm on

X

L(X,Y).

If

operator norm of an element

sup{llTxll

If

X

and

by t h e b i l i n e a r the linear d e n o t e d by X

: Y

Ilxll

and

T E L(X,Y)

~

Y

o(X,Y),

o{X,Y)

([79]

Other notations

X

into

are normed spaces, the

is defined by

w i t h one a n o t h e r

t h e w e a k e s t t o p o l o g y on

are continuous for all is referred

X

y E Y

such that is

t o a s t h e weak t o p o l o g y on

8.2).

and t e r m i n o l o g y employed in t h e n o t e s i s

Specifically,

Y

I}.

<, >,

x ÷ <x, y>

Y

X

a r e two v e c t o r s p a c e s in d u a l i t y

pairing

maps

i n d u c e d by

standard.

for

X.

will be denoted by

=

II II

is a normed space, we write

The space of all continuous linear operators from

IITll

If it

we f o l l o w [381 f o r t h e most p a r t .

Finally, which w i l l Let

for

later

u s e , we r e c o r d a lemma o f Drewnowski ( [ 3 6 ] )

be u s e d a t s e v e r a l E

be an a l g e b r a

is a finitely

additive

additive

(exhaustive

disjoint

sequence

junctures

of subsets

set function, or strongly

{E i}

from

in the text. of a set

then

g

bounded)

if

E .

S.

If

g : E ÷ G

i s s a i d t o be s t r o n g l y lim g(E i) = 0

We h a v e t h e f o l l o w i n g

for each

result

due

t o Drewnowski.

Lemma 1. strongly from

Let

E

be a o - a l g e b r a .

additive

set functions

E , then there

~i i s c o u n t a b l y

Drewnowski s t a t e s

on t h e this

[ ]

strongly {Ekj}

~ : E * G~

additive

of

{Ej}

E0 ' g e n e r a t e d tive

on t h e

=

is a disjoint

{Ekj}

)

of

{Ej)

generated

for a single and Uhl [34]

and by

o-algebra

[gil/(l

by

{Ekj}.

1.6),

additive

but t h e lemma

i n t h e f o l l o w i n g way: Equip

G~N w i t h

Igil

+ [gi[)2 i i s t h e "norm" o f

~(E) = ( ~ I ( E ) ,

~

g2(E) ....

is countably

Then e a c h

E0 .

such that

strongly

so by D r e w n o w s k i ' s lemma, t h e r e

{Ekj).

sequence

by

E i=1

such that by

{Ej}

is a sequence of

G-valued sequences.

defined

g = ( g 1 ' g2 . . . .

Now d e f i n e

: E * G

from Drewnowski's result

Ig[ where

result

G~N be t h e s p a c e o f a l l

the quasi-norm

~i

a-algebra

(see also Diestel

a b o v e c a n be d e r i v e d let

and

is a subsequence

additive

m e a s u r e in [36]

If

~i

gi )

in

G .

Then

~

is

is a subsequence

additive

is clearly

on t h e a - a l g e b r a , countably

addi-

2.

Basic Matrix Results

In t h i s matrices

section

we e s t a b l i s h

which will

be u s e d t h r o u g h o u t

i s a v e r y s i m p l e and e l e m e n t a r y real

numbers.

type result results

t h e two b a s i c

This result

for matrices

the sequel.

result

Lemma 1.

in t h e i r

Let

xij

on i n f i n i t e

The f i r s t

on m a t r i c e s

result

of non-negative

is then used to establish

a convergence

w i t h e l e m e n t s in a t o p o l o g i c a l

a r e o f an e l e m e n t a r y c h a r a c t e r

techniques

results

and r e q u i r e

group.

Both

only elementary

proofs.

~ 0

and

eij

> 0

for

i,j

E N.

If

lim xij

= 0

1

for each

j

and

lim xij 3

quence

{mi ) o f p o s i t i v e

Proof:

Put

and

mI = I .

xmmI < e21

for

Lemma 1 w i l l principle

integers

m ) m2.

(I)

m2 > mI

Then t h e r e

be u s e d d i r e c t l y

application

and

= xj

is a subse-

Xmimj < e i j

such that i s an

for

i ~ j.

xml m < e l 2

m3 > m2

xmm2 < e32

in several

o f Lemma 1 w i l l

such that

for

exists

later

m ) m3.

results

be t o e s t a b l i s h

but the

the basic

below.

Let

E

be a normed g r o u p and

Suppose

lim xij 1

then there

completes the proof.

B a s i c M a t r i x Theorem 2. E N.

i,

such that

xmmI < e31

matrix convergence result

i,j

for each

T h e r e i s an

xml m ~ e 1 3 , xm2m < e 2 3 , An e a s y i n d u c t i o n

= 0

for each

j

and

xij E E

for

(II)

f o r e a c h subsequence {m3}

Then

such t h a t

lim x i j = x3 i

In p a r t i c u l a r , Proof:

{ ~ Xin j} j=1

such t h a t

sume

k i = i.

there is a

sup {Xki j - x3[ • 6. J Set

there is

{xij - x j{ < 6

iI = 1

for

and

and n o t e t h a t {i k}

[ XikJk - Xik+lJk { • 6.

Set

(I)

6 • 0

convenience as{x i l j I - x31{ > 6.

{XilJl - x i 2 J l { • 6

1 ~ j ~ Jl" J2 • J1"

and

and a subsequence

Jl such t h a t

such t h a t

i • i2

we o b t a i n s u b s e q u e n c e s

{jk }

Now p i c k

and J2

such t h a t

C o n t i n u i n g by i n d u c t i o n , such t h a t

Zk£ = XikJt " - Xik+lJ£

and n o t e

[ Zkk [ • 6. Consider the matrix

matrix converge to verge to such that

0

0.

[{Zk£ [] = Z.

By (1), the columns of this

By (II), the rows of the matrix

so the same holds for the matrix ~ eij < ~. 13

]Zmkm£ [ < ekZ

Z.

Let

[xij] coneij • 0

By Lemma i, there is a subsequence for

{m k}

k # £.

By (If) there is a subsequence

{n k)

of

(m k}

such that

lim £=I~ Znkn£ = 0.

Then (3)

of

j.

For n o t a t i o n a l

and p i c k

i2 • i I

{xi2J2- x j 2 { • 6

(2)

{nj)

i s Cauchy.

uniformly with respect to

If the conclusion fails,

By ( I )

t h e r e i s a subsequence

lim x i i -- O. i

(k i}

that

(m3)

[ Znknk{ ~ [A#k )- Znknj~ { + [£=i )- Znknz[ < Z#k ~- e nkn~"

be such

9

~=1 Znkng~ [ " Now t h e f i r s t

t e r m on t h e r i g h t

hand s i d e o f (3) g o e s t o

by t h e c o n v e r g e n c e o f t h e s e r i e s by ( 2 ) .

But t h i s

contradicts

~" ek~"

0

as

k ÷

and t h e s e c o n d t e r m g o e s t o

(1) and e s t a b l i s h e s

the first

0

part of the

conclusion.

The u n i f o r m c o n v e r g e n c e o f t h e l i m i t ,

lim x i j

= xj

and t h e f a c t

1

that

limj x i j

exists

= 0

for each

and i s e q u a l t o

lim i xii

O.

implies that

In p a r t i c u l a r ,

the double limit

this

A matrix

Theorem 2 w i l l

will

[xij]

be c a l l e d

terminology will

implies

be t h e b a s i c t o o l used t h r o u g h o u t t h e

which s a t i s f i e s a

R matrix

be i n d i c a t e d

conditions

forms ( [ 2 ] ,

character

zero.

In f a c t ,

i f one f i r s t

converges to zero, fact

the diagonal

[53]).

then it

sequence

shows o n l y t h a t i s not d i f f i c u l t

than the

t h e h y p o t h e s i s and t h e c o n However, Theorem 2 can

a l s o be viewed as a d i a g o n a l t h e o r e m in t h e s e n s e t h a t o f Theorem 2 imply t h a t

of

in t h e n e x t s e c t i o n ) .

A n t o s i k - M i k u s i n s k i D i a g o n a l Theorem in t h a t have v e r y d i f f e r e n t

( I ) and ( I I )

(the reason f o r the use of t h i s

The B a s i c M a t r i x Theorem 2 has a v e r y d i f f e r e n t

clusions

limij x i j

= O.

This matrix result sequel.

i

{xii}

the hypotheses converges to

the diagonal to use t h i s

sequence

t o show t h a t

in

t h e columns o f t h e m a t r i x a r e u n i f o r m l y c o n v e r g e n t . Matrix results

established

in [6] -

of a very similar [11] and [ 7 3 ] .

have been u s e d t o t r e a t and f u n c t i o n a l treated

analysis.

in c h a p t e r s

n a t u r e t o Theorem 2 have been The m a t r i x r e s u l t s

a wide v a r i e t y

of topics

of these papers

in both measure t h e o r y

Much o f t h e c o n t e n t o f t h e s e p a p e r s w i l l

4, 5, 8 and 9.

I t s h o u l d be p o i n t e d o u t t h a t

the functional

analysis

text

of

be

10 E. Pap ( [ 5 6 ] ) systematic al analysis

uses the Antosik-Mikusinski

manner t o t r e a t and i n t h i s

notes except that

a variety

D i a g o n a l Theorem i n a

of classical

topics

s e n s e i s v e r y much i n t h e s p i r i t

we s y s t e m a t i c a l l y

in functionof these

e m p l o y t h e B a s i c M a t r i x Lemma.

3.

~

Convergence

In t h i s quence.

section

we i n t r o d u c e

T h i s n o t i o n was i n t r o d u c e d

further

explored

i n [71 - [ 1 1 ] ;

Boundedness Principle The

the notion of a

" K " in the description

convergent

se-

by P. A n t o s i k i n [6] and was

further

and b i l i n e a r

~

applications

to the Uniform

maps a r e g i v e n i n [14]

below is

and [751.

i n h o n o r o f t h e members o f t h e

Katowice Branch of the Mathematics Institute

o f t h e P o l i s h Academy o f

Sciences who have extensively studied and developed many of the results pertainin 8 to

X convergent sequences.

As a h i s t o r i c a l note, i t should be pointed out that S. Mazur and W. Orlicz introduced a concept very closely related to that of a convergent sequence in [52), Axiom I f , p. 169. They essentially introduced the notion of a

~

(metric linear) space which is defined

below and noted that the classical Uniform Boundedness Principle holds in such spaces.

A. Alexiewicz also studied consequences of

this notion in convergence spaces ([i] axiom A½ , p. 203). should also be noted however that the notion of a s e q u e n c e and t h a t versions linear

spaces

Definition E

in the classical

1.

is a

Let T- R

(E,T)

permits

{ .

If the topology r- ~

Xxk

be a t o p o l o g i c a l

}

convergent

the formulation in arbitrary

of

metric

to the situation

Uniform Boundedness Principle.

convergent

gent to an element

of

bounded s e t

(Theorem 4 . 2 b e l o w ) i n c o n t r a s t

has a subsequence

tion

K

of the Uniform Boundedness Principle

encountered

in

of a

~

It

group.

A sequence

sequence if each subsequence of

such that

the series

E

.

is

k Xxk

{x i} {x i )

r-conver-

x E E.

•

is

convergence.

understood,

we d r o p t h e

r

in

the descrip-

12 Note t h a t to

0

any

v- R

convergent sequence

by t h e Urysohn p r o p e r t y ,

i.e.,

subsequence which converges to

0.

(x i }

is

v-convergent

any s u b s e q u e n c e o f

{x i }

has a

In c o m p l e t e s p a c e s t h e c o n v e r s e

holds. Notice that

in t h e B a s i c M a t r i x Theorem 2 . 2 ,

t h e rows o f t h e m a t r i x a r e

sense).

in

Let

E.

E

H k [Xik[ < ~"

converges in

convergent

iff

R

E.

Thus,

the statement

matrix.

{x i }

converge to

has a subsequence

The c o m p l e t e n e s s

implies that

{Xik}

the series

in c o m p l e t e s p a c e s a s e q u e n c e i s

it converges to

In g e n e r a l

{x i }

implies

( i n some u n i f o r m

be a c o m p l e t e normed g r o u p and

Then any s u b s e q u e n c e o f

such that Xlk.

convergent

This is the reason for the terminology:

Example 2. 0

K

assumption (II)

~

0. i n Example 2 i s f a l s e

as the following

e x a m p l e shows.

Example 3.

Let

such that ek and

tj

Coo

= 0

be t h e v e c t o r

eventually.

be t h e s e q u e n c e in O

elsewhere.

Equip

which has a

0

in

Coo

0

b u t i s not

K

g

it

has the property

convergent.

that

There are,

1

in the {(1/j)ej}

Let

in

Coo.

This

of the series

That is,

this

sequence

convergent.

a (normed) s p a c e i s c o m p l e t e 0

is

h o w e v e r , normed s p a c e s w h i c h h a v e t h i s

b u t a r e not c o m p l e t e .

property

that

is called

a

A topological

any s e q u e n c e w h i c h c o n v e r g e s t o space.

{tj}

kth coordinate

e v e r y sequence which c o n v e r g e s to

property

K

sequences

with the sup-norm.

Coo.

E x a m p l e s 2 and 3 m i g h t s u g g e s t t h a t iff

real

but no s u b s e r i e s

c o n v e r g e s t o an e l e m e n t o f

converges to

Coo

Consider the sequence

sequence converges to ~(l/j)ej

Coo

space of all

Klis

([45])

group which has the 0

is

X

convergent

h a s g i v e n an e x a m p l e o f a normed

13 space which is a

space but is not complete. (See also [261,

~

Theorem 2 and [48] Theorem I . )

The example given by Klis is f a i r l y

involved and we do not present i t here since i t is not needed in the sequel. The notion of a

K space was originally introduced in another

equivalent form by S. Mazur and W. Orlicz in [52], Axiom I I , p. 169, where i t was observed that the classical Uniform Boundedness Principle holds for such spaces.

A. Alexiewicz also studied this notion in

I l l , Axiom A½ , p. 203. We record here several other interesting properties of spaces and give references to the l i t e r a t u r e where these results can be found.

We do not present the results here since they are not

germane to the theme of these lecture notes. F i r s t , any metric group which is a space ([26] Theorem 2).

~

space is also a Baire

More generally, any Frechet topological

group is also a Baire space ([26] Theorem 2). Baire spaces which are not

K spaces.

Second, there exist

Indeed, Burzyk, Klis and

Lipecki have shown that any i n f i n i t e dimensional F-space contains a subspace which is a Baire space but is not a

~

space ([26] Theorem

3). Recall that a subset

B of a topological vector space is

bounded i f f for each sequence (xj} c B and each sequence of scalars (tj)

which converges to

0,

the sequence (tjxj}

Using t h i s c r i t e r i a and the notion of duce the concept of a

Definition 4. Bc E

is T- ~

Let

is

0.

X convergence) we can i n t r o -

X bounded set.

(E,T)

be a topological vector space.

A subset

bounded i f f for each sequence {xj} c B and each

sequence of real numbers {tj) (tjxj}

converges to

T- ~

convergent.

s u c h that

lim t j = 0,

the sequence

14 Since a bounded s e t sequence i s not

K

sequence converges to

i s a l w a y s bounded.

{ej}

is

~

The c o n v e r s e i s f a l s e ;

i n Example 3.

useful,

if follows

hess principle difficulties convergent

the

Coo

but

By t h e o b s e r v a t i o n

in an F - s p a c e a s u b s e t

the notion of

in r e f o r m u l a t i n g

(Corollary associated

4.4).

K

i s bounded i f f

convergent

Let

mo

{tj

: j E ~}

each

k.

D e f i n e a norm ( i n d u c e d by

Z ej

is

is finite.

Pick

and,

{sj} E c o

{{-subseries

where

therefore,

and t h e

be

(sj

~

real

bounded.

on

convergent

CE

{ej}

sequences

{~k } E ~1 {~k })

n [I j=l Z e k j - C{k j : j E

E),

boundedness.

A

s p a c e i s a l w a y s bounded;

Consider the sequence

{{

for each subsequence,

~

some a n n o y i n g

We

phenomena in t h e e x a m p l e b e l o w .

be t h e s p a c e o f a l l

= H {~jtj{. J

uniform bounded-

however,

vector

sequence needn't

that

{{(tj}[{

boundedness is quite

the classical

with the notion of

an e x a m p l e o f t h i s

Example 5.

K

There are,

sequence in a topological

however, a

series

K

consider

T h i s s e q u e n c e i s bounded i w

that

be n o t e d l a t e r

particularly

present

any

bounded.

As w i l l

of

0,

~ bounded ( b y t h e a r g u m e n t in Example 3 ) .

in Example 2, it

convergent

~){{

with mo

~k $ 0

in

mo.

m0

= j=Z n+ll~kj

: j E ~}

I{ ] I - ~

{÷

The

0

convergent

are distinct,

for

(because

denotes the characteristic is

such

by

(ej) in

(tj}

function in

mo.

t h e n no s u b s e r i e s

eo

)" s e j=l kj kj

is

{{

{ { - c o n v e r g e n t t o an e l e m e n t o f

mo

(indeed

¢O

Z sk ek

j=t

Hence,

converges coordinatewise ( e j}

is

t o an e l e m e n t o f

[{ { [ - ~K c o n v e r g e n t b u t not

~

\ mo)-

{{ {{- ~K bounded.

If

15 The construction in Example 5 can be generalized to give a large number of examples of bounded. Let

E

~

convergent sequences which are not

be a o-algebra of subsets of a set

S.

Let

v : H ÷ R be a positive, bounded, f i n i t e l y additive set function. If

S(H)

is the space of all

semi-norm on

S(E)

by

disjoint sequence from where

CE

E-simple functions, then

II~IIv [

=

v

Thus,

~}~Idv.

=

Let

and consider the sequence

denotes the characteristic function of

Lemma 1.1, any subsequence of that

I1~11

(Ej}

v

induces a

be any

(Ej) (CE } 3

E.

has a subsequence

in

By Drewnowski's (Ejk}

is countably additive on the o-algebra generated by

(CEj}__ is II II- ~

by countable additivity).

(tj} £ c O and the

are distinct, then no subseries of

~ tic E J

an element of

is

S(H).

Thus,

(~

}

such {Ejk}.

n II Z CE. - CUE i [I ÷ 0 K=i Jk ~k

convergent (since However, if

S(H),

is

(tj : j E ~}

il II- ~ convergent to

II II- ~

convergent but not

J I] II- ~

bounded.

This construction actually shows that the Drewnowski Lemma can be viewed as a result concerning is a disjoint sequence from (CE } J vi H

is

II IIv- ~

E,

~

convergence.

Namely,

if

{Ej}

then the sequence

convergent for any

v.

More

generally, if

is a sequence of positive, bounded, finitely additive measures on and

T

is the metric topology on

of semi-norms

( II I IVl. },

then

(CE3}

Z(H)

induced by the sequence

T- ~

is

convergent.

In the remainder of this section we collect some miscellaneous results concerning

~

convergence which, hopefully, will illustrate

some of the applications of the notion of

K

convergence.

First, it is well known that in the dual of a normed space

X,

16

a s e t may be weak ~ bounded but not norm bounded. dual,

~1,

of

Coo,

norm bounded.

the sequence

(jej)

However, f o r weak )(- R

For example,

in t h e

i s weak X bounded but not

bounded s e t s we have t h e f o l -

lowing r e s u l t . t

Theorem 6.

Let

bounded, t h e n X

X B

be a normed s p a c e .

B c X

i s weak ~-

i s norm bounded ( a n d , h e n c e , norm- ~

bounded s i n c e

is complete).

Proof:

Let

(xj) c B

converges to

O.

and

{tj}

For each

,

and

If

j

be a s e q u e n c e o f s c a l a r s

pick

xj E X

such t h a t

which

IlxjII

= 1

t

l<xj,xj>l ) llxjll

- l/j.

Consider the matrix

i

[zijl

= [
4Jtilxi>].

Since

Jltilx i ÷ 0

in norm and

I

(4ltj[xj}

i s weak t~- K c o n v e r g e n t ,

the matrix

[zijl

is a

~

matrix.

By t h e B a s i c M a t r i x Theorem, lim z i i i Thus, B

= lim [ t i ] < x i , x i > i

= 0

so t h a t

lim ] t i ] i

]]xilJ

= O.

i s norm bounded.

The p r o o f above a l s o shows t h a t t h e ( u n d e s i r a b l e ) p l a y e d in Example 5 d o e s not o c c u r in weak ~ t o p o l o g i e s .

phenomenon d i s That i s ,

the

0

p r o o f above shows t h a t

a weak ~- ~

convergent sequence

{xj)

i s norm

0

bounded and s i n c e

X

is complete, the sequence

bounded and, t h e r e f o r e ,

weak ~- ~

Another interesting

infinite

~

convergence concerns the

Again i t

i s w e l l known t h a t

in

d i m e n s i o n a l normed s p a c e s a s e q u e n c e which c o n v e r g e s t o

weakly w i l l

generally

not be norm c o n v e r g e n t .

exception to t h i s statement.) co

i s a l s o norm-

bounded.

property of

weak t o p o l o g y i n a normed s p a c e .

{xj)

i s weakly convergent to

For weak- ~

0

(£1

0

i s an i n t e r e s t i n g

For example, the sequence { e j )

in

but i s c e r t a i n l y not norm convergent.

convergent sequences, we, however, have

17 Theorem 7.

Let

convergent,

then

Proof: the

X

be a normed s p a c e .

{xj)

By r e p l a c i n g

{xj),

converges to

X

0

X

{xj} x

For each

j

pick

and

<xj,xj> = }{xjl{.

By t h e Banach-

)

e

t o an e l e m e n t

s u b s p a c e g e n e r a t e d by

e

= 1

e

Alaogu Theorem,

i s weak- X

in norm.

is separable.

e

{Ixjll

such that

(xj} ~ X

by t h e c l o s e d l i n e a r

we may assume t h a t

*

x j fi X

If

has a s u b s e q u e n c e

{Xkj}

w h i c h c o n v e r g e s weak ~

e

e

E X .

Consider the matrix

[zij]

= [<Xki,Xkj>]..

e

By t h e weak ~ c o n v e r g e n c e o f {xjt,

[zij]

lim z i i

is a

R

matrix.

= lim [ [ X k i [ [

t o any s u b s e q u e n c e o f lim

{Xkj}

= 0.

and t h e weak- ~

By t h e B a s i c M a t r i x Theorem,

S i n c e t h e same argument can be a p p l i e d

{xi},

t h e Urysohn p r o p e r t y i m p l i e s t h a t

llxill ffi O. Actually,

Theorem 7 can be improved t o c o n c l u d e t h a t a s e q u e n c e

i n a normed s p a c e i s weak- ~ c o n v e r g e n t i f f That i s ,

8.

Let

X

be a normed s p a c e .

weak- ~ c o n v e r g e n t i f f

Proof:

it

i s norm- ~ c o n v e r g e n t .

we have

Corollary

Clearly

Suppose

{{yn{ { ~

it

Then a s e q u e n c e in

is

i s norm- ~ c o n v e r g e n t .

{x n}

I / 2 n.

i s weak- ~ c o n v e r g e n t . 0.

Let Next,

{yn } let

By Theorem 7,

be a s u b s e q u e n c e o f {z n}

~ z n = z, n=l

where t h e c o n v e r g e n c e i s in t h e weak t o p o l o g y .

But,

{x n}

{x n}

be a subsequence o f

z E X be such that

(1)

X

norm- ~ convergence i m p l i e s weak- ~ c o n v e r g e n c e .

norm c o n v e r g e n t t o that

convergence of

is

such {yn }

and

18 n

so t h e s e r i e s respect

Zz i

~

Z 1/2 i i=m

i s norm Cauchy.

Consequently,

(1) h o l d s w i t h

t o t h e norm t o p o l o g y .

We i n d i c a t e

an a p p l i c a t i o n

derive the classical Zx i

n

Ilzill

Z i=m

Orlicz-Pettis

i n a normed s p a c e

each subseries

o f Theorem 7 by u s i n g Theorem 7 t o

EXk.

X

Theorem.

Recall that

i s norm ( w e a k l y ) s u b s e r i e s

i s norm ( w e a k l y ) c o n v e r g e n t i n

a series

convergent if X.

The

i

remarkable Orlicz-Pettis

Theorem 9.

Let

c o n v e r g e n t in

Proof: [30]

X X,

Theorem ( [ 5 5 1 ,

be a normed s p a c e . then

It suffices

Ex i

7).

But

If

i s g i v e n by

Zx i

i s norm s u b s e r i e s

[[xill

t o show t h a t

IV. 1 . 1 ; O r l i c z - P e t t i s

l e n g t h in s e c t i o n

[59])

results Ix i}

÷ 0

i s weak s u b s e r i e s convergent.

(See Theorem 7.1 o r

a r e d i s c u s s e d a t some l e n g t h in is clearly

weak- K c o n v e r g e n t so

Theorem 7 g i v e s t h e r e s u l t .

Note t h a t logy.

t h e a n a l o g u e o f Theorem 7 i s f a l s e

F o r example,

but i s c e r t a i n l y

the sequence

{ej}

not norm c o n v e r g e n t .

f o r t h e weak ~ t o p o -

i s weak ~ - K c o n v e r g e n t in

£~

C o n c e r n i n g t h e weak ~ t o p o l o g y ,

we do, however, have

Theorem 10.

Suppose

X

i s a normed s p a c e w h i c h c o n t a i n s 0

isomorphic (topologically) vergent,

Proof:

then

Pick

(xj)

to

gl.

converges to

×j 6 X

such t h a t

If

no s u b s p a c e

i

{xj} _c X

0

in norm.

I Ixjll

= 1

i s weak ~ - ~ c o n -

0

and

<xj,xj> + (I/j)

>

i

l lxjl[.

Since

X

contains

no s u b s p a c e i s o m o r p h i c t o

£1,

the se-

19

quence

(xj}

([49]

2.e.S).

that

(Xk.} J

contains

a subsequence

Consider the matrix

(Xkj} zij

w h i c h i s weak Cauchy

= <Xk ,Xk.>. j 1

From t h e f a c t s

e

follows

i s weak Cauchy and

that

[zii]

is a

~

{xj}

i s weak ~ - ~ c o n v e r g e n t ,

matrix.

it

The B a s i c M a t r i x Theorem ime

plies

that

limi z i i

= 0

so t h a t

lim I]Xkill

= 0.

S i n c e t h e same

t

a r g u m e n t can be a p p l i e d

t o any s u b s e q u e n c e o f

(xi),

the Urysohn

w

property

implies

that

This result

i s somewhat a n a l a g o u s t o t h e D i e s t e l - F a i r e s

on weak ~ s u b s e r i e s It

l i m { i x i ] I = 0.

convergence ([33]

Theorem

o r Theorem 1 0 . 1 0 ) .

i s not known i f t h e c o n v e r s e o f Theorem 10 h o l d s .

We conclude this section by establishing a result due to E. Pap concerning the boundedness of adjoint operators (private communication).

Recall t h a t i f

X and

Y are normed spaces and

linear operator with domain, D(T), a dense subspace of

T

is a

X and range

e

in

Y,

then the adjoint operator i

T

t

the domain, D(T ),

of

is defined by the following:

e

T

is

~

~

continuous on

D(T)} and

T

D(T)

Thus, for

be dense in x 6 D(T), y

6 Y

0

: y T

is

t

0

: D(T ) ÷ X

the unique continuous extension of that

~

D(T ) = (y

y T

is defined by

to

X.

r

Ty

is

( I t is necessary

X for this definition to make sense.) 6 D(T ), we have

= ; t h i s

formula agrees with the usual definition of the adjoint of a bounded linear operator.

It can happen, when T

is not continuous, that

e

D(T )

consists of only the

operator, then

D(T°)

0

vector; however, i f

is weak~ dense in

Y'.

T

is a closed

Concerningthe adjoint

operator, we have the following interesting result of E. Pap.

Theorem 11.

Let

X

be a

~ - s p a c e and

|

Then

T

T : X ÷ Y I

i s a bounded l i n e a r

operator

on

D(T ).

be l i n e a r .

20 Proof:

It suffices 0

then

{T Yi }

,

t o show t h a t

i s bounded.

!

!

[]Yi

I] ~ I ,

Pick

xi E X

such that

]Ixi[ I = 1

and

Thus, i t s u f f i c e s

t o show t h a t

l

{ } {t i }

i s bounded.

is a positive

{t i }

zij that

[zij]

converges to

: is a

t o show t h a t

which c o n v e r g e s t o

O,

if then

O. 0

× j > ( : <J

~ - matrix.

First

iYi '

for each

We show i,

0

= l i m <4~iT Yi , 4 ~ x j >

J

it suffices

i

T Yi ' ,

lim z i j

For t h i s ,

sequence of scalars g

j,

and

!

J]T Yi II ~ + i . ,

i f {Yi } ~ D(T )

u

= 0

since

J~

= 0

since

x j ÷ O.

Next, for each

J lim z i j i

4t-~ix j * 0 such that x E X.

= lim i and

X

is a

the subseries

K - s p a c e so t h e r e ~j J~kkjXkj

tV~iy i

* O.

Now

is a subsequence

{kj}

c o n v e r g e s in norm t o an e l e m e n t

Then, )-j• Z i k j = )- < J-t -i T' Yi' 4tkkjXkj> t

i

= <J~ii T Yi'X> t

= ÷ 0 since

Jt-~ly i

* 0 .

Thus,

[zij]

is a

K-matrix.

4.

The U n i f o r m B o u n d e d n e s s P r i n c i p l e

In t h i s Principle trix

section

(UBP).

the well-known Uniform Boundedness

We g i v e a p r o o f o f t h e UBP b a s e d on t h e B a s i c Ma-

Theorem ( 2 . 2 )

is so often

we d i s c u s s

instead

given (cf.

of the familiar

for example,

i n g m a t r i x m e t h o d s we a r e a c t u a l l y UBP w h i c h i s v a l i d the classical to indicate

UBP f o r F - s p a c e s

we d e r i v e

motivate

state

Let

bounded on

X, t h e n

subsets o f

X.

X

The c o n c l u s i o n

a version

of the

UBP f o r normed s p a c e s in o r d e r

be a B - s p a c e . {T i}

If

is pointwise

i s u n i f o r m l y bounded on t h e

norm o f

This conclusion

clearly

i s bounded,

]]

]]-bounded

o f t h e U~P i s u s u a l l y

i s bounded, where

: { l ] T i l I}

to

UBP.

{T i} c L(X,Y)

o f t h e norm v e r s i o n

B c X

In order

UBP.

of the general

in the form

if

able to establish

By e m p l o y -

o f t h e Nikodym B o u n d e d n e s s Theorem o f

the classical

Theorem I .

rem 1 s i n c e

II.1.11).

UBP c a n be u s e d i n t h e a b s e n c e o f com-

a version

the discussion

T.

[38]

a s an i m m e d i a t e c o r o l l a r y .

measure theory from the general We f i r s t

[19],

w i t h n_qocompleteness a s s u m p t i o n s and w h i c h y i e l d s

how t h e g e n e r a l

pleteness,

Baire Category proof which

I]TII

written

is the operator

implies the conclusion

in Theo-

then

llTixll ~ sup llTill sup llxll i

for each It

i

and

x 6 B.

is well-known that

the conclusion

the completeness a s s u m p t i o n on quence of linear = it i

B

functionals

is pointwise

X {fi )

i n Theorem 1 i s f a l s e

is dropped. defined

on

if

F o r example, t h e s e Coo

by

< fi'

{tj} >

bounded b u t not u n i f o r m l y bounded on t h e bounded

22

subset

: j E N)

{ej

coo.

of

Thus, i f one i s t o s e e k a v e r s i o n

o f t h e UBP w h i c h i s v a l i d

arbitrary

normed s p a c e s

X, t h e f a m i l y o f bounded s u b s e t s

too large

to insure

a pointwise

that

pointwise

of

X

be as l a r g e

that

a

this

and i t would be o f i n t e r e s t

family should

to find a family

of s e t s which c o i n c i d e s

w i t h t h e f a m i l y o f bounded ~ e t s when t h e

space

I t i s shown in C o r o l l a r y

X

is complete.

bounded s e t s We f i r s t

has this establish

thus,

and o n l y i f

our general

a sequence

Txj * 0

t i o n we w r i t e

4 that

the family of

property.

be t h e w e a k e s t t o p o l o g y on tinuous;

is

i s u n i f o r m l y bounded on e a c h

In o r d e r t o be i n t e r e s t i n g

as possible,

is

I t would be d e s i r -

with the property

bounded s e q u e n c e o f o p e r a t o r s

member o f t h e f a m i l y .

X

bounded s e q u e n c e o f o p e r a t o r s

u n i f o r m l y bounded on e a c h member o f t h e f a m i l y . able to have a family of subsets

of

for

X

If

such that

{xj}

in

for each

F - K bounded

UBP.

X

F ~ L(X,Y),

e a c h member o f converges to

T fi F.

let F

0

is con-

in

In o r d e r t o s h o r t e n

(F- ~ convergent}

for

T(F)

T(F)

if

the nota-

T ( F ) - R bounded

(T(F)- ~ convergent).

Theorem 2.

Let

be s u c h t h a t

X,Y

{Tix}

be m e t r i c

linear

s p a c e s and l e t

i s bounded f o r e a c h

x E X.

T i E L(X,Y)

Then

(i)

{T i )

i s u n i f o r m l y bounded on

{Ti}- ~ convergent

sequences

(ii)

{T i}

i s u n i f o r m l y bounded on

{ T i } - K bounded s e t s .

and

Proof: in

Y

If (i) and a

is false,

is a balanced neighborhood

(Ti}- K convergent sequence

not a b s o r b e d by such that

there

U.

Thus,

TmlXnl ~ U.

h e s s and t h e f a c t

that

Set

there

exists

k 1 = 1.

lim Tix j = 0 J

(xj}

positive

U

such that integers

i,

there

0

{Tix j ) m1

Then by t h e p g i n t w i s e for each

of

and

bounded-

exists

is n1

23 k2 > k I

~,

such that

{Tix j

1 ~ j ~ n I} c k 2 U.

such that

: 1 ( i 6 mI ,

1 ( j < -} U {Tix j

By assumption, there exist

Tm2Xn2~ k 2 U.

Note that

m2 • m I

and

ing this construction produces subsequences such that

Tmixni ~ kiU.

If

[tiT m x n ].

and

n2 • n I.

{mi},

t i = I/k i, then

Now consider the matrix

m2

: 1 ( i <

{n i}

n2 Continu-

and

{k i)

{t i) 6 c o .

By the pointwise bounded-

ij ness assumption and the this is a Thus,

K

matrix.

{Ti}- ~ convergence of the sequence Hence, by Theorem 2.2,

t i TmiXni 6 U

for large

i.

{xj),

liml tiTmiXni = O.

This contradicts the construction

and establishes (i). For

(ii), let

B

be

{Ti}- X bounded.

To show

bounded, as in part (i), it suffices to show each

{x i} c B.

Let

{t i) 6 c o .

convergent so part (i) implies

Then

{Tix i}

{41til x i}

{Ti(JItil xi)}

lim Jltil Ti(41til x i) = lim ItiITix i = 0 i i

{TiB}

is bounded for is

{Ti}-

is bounded.

so that

is

Hence,

{Tix i} is bounded.

Note that the conclusion in Theorem 2 is much sharper than the conclusion of the classical UBP in the sense that the family of subsets of

X

where the given sequence

{T i }

is uniformly bounded

depends very much on the particular sequence sequence

{Ti}, the family of

(It

is easy

normed space

X

of continuous

linear

member o f t h e

family,

set.

if

Indeed,

to check that

has the

property

functionals then

{x i }

is

Indeed, given a

{Ti}- X convergent sequences

bounded sets) may even contain subsets of bounded.

{Ti}.

if that

on

X

X

which are not norm-

a family

of subsets

any pointwise is

uniformly

e a c h member o f t h e

({Ti}-

family

an u n b o u n d e d s e q u e n c e

of a

bounded sequence b o u n d e d on e a c h is

a bounded

in a normed space

24 e

X, p i c k

o

t

xi 6 X

such that

o

<xi'

xi> = [[×il[

and

[Ix ill

= 1.

Then

I

(x i}

i s u n i f o r m l y bounded on bounded s u b s e t s

f o r m l y bounded on t h e s e q u e n c e

{xi}. )

of

X

but is not uni-

For example, the sequence

I

{i e 2 i }

in

£1 = (Coo)

is pointwise

bounded on

u n i f o r m l y bounded on t h e unbounded s u b s e t

Coo

{j e 2 j + l

and i s a l s o

: j £ ~}

of

Coo"

In o r d e r t o o b t a i n perry that

the largest

each pointwise

3.

(i)

Let

{T i}

family

~9 w i t h t h e p r o -

bounded s e q u e n c e o f o p e r a t o r s

bounded on e a c h member o f

Corollary

possible

~,

we u s e t h e

{T i } c L(X,Y)

T(L(X,Y))

be p o i n t w i s e

i s u n i f o r m l y bounded on

is uniformly topology of

bounded.

X.

Then

L(X,Y)- K convergent

se-

q u e n c e s and (ii)

{T i}

i s u n i f e ~ m l y bounded on

It is worthwhile noting that quences

( ~ bounded s e t s )

n o t t h e same. different.

That is,

in

Lemma.

However, {e i }

tive

measure

g

and

g({nk})

= 0

on

([41]

g(R) = 1

but

convergent family of

integers,

for e~ch

(n k : k £ ~) = R. P

ba- K

on t h e power s e t

can be shown a s f o l l o w s : onto

{v i}

i s not

of positive

k.

sequences

K

Let Let

= 0

?

in

f o

families is

se-

c a n be

{ui }- ~

con-

by D r e w n o w s k i ' s

c o n v e r g e n t s i n c e g i v e n any s u b there of

exists

~

a finitely

such that

be a 0 - 1 f i n i t e l y Define

g

by

k.)

i n Theorem 2 i s ,

addi-

~(~{nk})

= 1

of such a measure

be t h e o n e - o n e map

for each

L(X,Y)- K convergent

these

ba = mo

convergent

3 a r e in g e n e r a l

{e i}

(The e x i s t e n c e

p. 358, 20. 3 8 ) . g((nk})

topologies

mo t h e s e q u e ~ c e

f o r any g i v e n s e q u e n c e

{n k}

of

in Theorem 2 and C o r o l l a r y

vergent

sequence

the families

for certain

For example,

L ( X , Y ) - [ bounded s e t s .

k ÷ nk additive

g(A) = o ( f - l ( A ) ) .

Thus,

the family of

in g e n e r a l ,

s e q u e n c e in C o r o l l a r y

larger 3.

from measure Then (Ti}-

than the

25 Since the topology of

Corollary (i)

T(L(X,Y))

topology of

X, we o b t a i n

immediately.

4.

Let

{T i }

{T i } ~ L(X,Y)

i s weaker t h a n t h e n a t u r a l

X

be p o i n t w i s e bounded.

Then

i s u n i f o r m l y bounded on

II- ~ convergent sequences

i s u n i f o r m l y bounded on

I I - K bounded s e t s .

and (ii)

{T i}

Since the families i n c i d e in

o f norm bounded s e t s

B-spaces, Corollary

and

4 in p a r t i c u l a r

B - s p a c e s g i v e n in Theorem 1 as a C o r o l l a r y . lary 4 yields

t h e UBP f o r

Orlicz

p.

([52],

~

and

169) ( r e c a l l

that

s h o u l d be n o t e d t h a t

~

trasted

contains

in a r b i t r a r y

metric

with the classical

g i v e n by Mazur and O r l i c z Corollary

t h e UBP f o r

More g e n e r a l l y ,

a metric

linear

Corol-

0

the notions of

space is a

is a X

linear

spaces.

~

K

sequence).

convergent sequence

bounded s e t a l l o w one t o f o r m u l a t e v e r s i o n s

are valid

bounded s e t s c o -

s p a c e s as p r e s e n t e d by Mazur and

space i f e v e r y sequence which c o n v e r g e s t o Again i t

K

o f t h e UBP w h i c h

T h i s s h o u l d be c o n -

UBP as w e l l as t h e v e r s i o n o f t h e UBP in [52].

4 now y i e l d s

immediately the classical

UBP f o r

F-

spaces.

Corollary

5.

Let

w i s e bounded. (i)

Proof:

be an F - s p a c e and l e t

{T i ) ~ L(X,Y)

be p o i n t -

Then

{T i}

(ii)

X

{T i}

i s u n i f o r m l y bounded on

ll-bounded subsets of

X

and

is equicontinuous.

Conclusion (i)

f o l l o w s from C o r o l l a r y

complete space the families

o f bounded s e t s

4 (ii)

and

K

s i n c e in a bounded s e t s

coincide. To e s t a b l i s h

(ii),

it

suffices

t o show t h a t

Ti× i ÷ 0

whenever

26

xi ÷ 0

in

scalars

X.

Let

{t i}

xi ÷ 0

such that

in

X and pick a sequence of p o s i t i v e

ti * ~

is bounded. Consequently,

and

t i x i * 0.

By ( i ) ,

( I / t i) Ti ( f i x i) = Tix i ÷ 0

{Ti(tixi)} since

{ l l t i } £ c o. The c o n c l u s i o n of the classical To i n d i c a t e

in ( i i )

i s t h e c o n c l u s i o n in t h e u s u a l

UBP f o r F - s p a c e s ( c f . an a p p l i c a t i o n

[38]

statement

II.1.11).

o f t h e g e n e r a l UBP in Theorem 2, we

d e r i v e a v e r s i o n o f t h e Nikodym Boundedness Theorem.

The c l a s s i c a l

Nikodym B o u n d e d n e s s Theorem i s g i v e n in t h e f o l l o w i n g s t a t e m e n t :

Theorem 6.

Let

~i

be c o u n t a b l y a d d i t i v e .

: Z ÷ R

E E Z, t h e n

E

be a

{gi(E)

u-algebra

: i C N,

of subsets of a set If

E E E}

{~i(E)}

S

and l e t

i s bounded f o r e a c h

i s bounded.

This r e s u l t is called "a s t r i k i n g improvement of the UBP" for the space

ca(Z)

by Dunford and Schwartz ([38] I I I . l . 5 ) .

We recast

t h i s theorem in a form which i s more analogous to the usual statement in the UBP and then indicate why the result does not follow from the c l a s s i c a l UBP. Let

S(Z)

be the space of a l l

E-simple functions equipped with

the sup-norm. Each ~i induces a continuous linear functional S(Z) via integration, variation of

~i'

[~i I '

< f i ' y> = ~Yd~i" The dual norm of on

S, i . e .

,

l l f i ] I = l~i](S).

f

f i on is the

This v a r i -

I

ation norm is, however, equivalent to the norm l l ~ i l l sup{[~i(E) I : E E Z}. form:

i f {fi }

=

In t h i s notation, Theorem 6 takes the familiar

is pointwise bounded

cn

S(Z),

then

{ l l f i l []

is

bounded. Note that t h i s r e s u l t is not obtainable from Theorem 1 since the space

S(Z)

is not complete (or even 2nd category [66]).

derive Theorem 6 from the general UBP given in Theorem 2.

We will

27

Actually, Darst ([34]) has shown that Theorem 6 is even valid for bounded, f i n i t e l y additive set functions, and, furthermore, i t is known t h a t

Theorem 6 i s a l s o v a l i d

a r e not a - a l g e b r a s

([67];

for certain

a l g e b r a s o f s e t s which

s e e a l s o Theorem 5 . 1 2 ) .

We w i l l

derive a

v e r s i o n o f t h e Nikodym B o u n d e d n e s s Theorem f o r bounded f i n i t e l y tive

set functions

addi-

on a - a l g e b r a s .

For t h e p r o o f o f t h e Nikodym Boundedness Theorem, we r e q u i r e

the

f o l l o w i n g " w e l l - k n o w n " lemma ( [ 3 7 ] ) .

Lemma 7. gi

Let

: ~ ~ R

E E ~}

~

be an a l g e b r a o f s u b s e t s o f a s e t

be bounded and f i n i t e l y

i s bounded i f and o n l y i f

for each disjoint

Proof: each

sequence

such that

{gi(Ej)

: i 6 ~,

there is a p a r t i t i o n

(E,F)

] ~i (E) I - [ ~i(S) ] > M.) of

Now e i t h e r

S

such that

an

i2 > i I

as

S

of

S

i s bounded

and an integer

i

(This follows since implies

I ~i( S \ E) I iI

and a p a r t i t i o n

Pick whichever of B1

and set

above to obtain a p a r t i t i o n

satisfying

: i E ~,

Note that for

sup ( l ~i (H N El ) ] : H 6 ~, i 6 ~} = ~

H 6 h, i 6 ~} = ~. {ij}

j E N}

let

min{ I ~il(E I) ], [ ~ii(F I) ] } > i .

s a t i s f i e s t h i s condition and label i t B1

{gi(E)

E 6 ~} = ~.

Hence, there exist

~i(H N FI) I : H 6 ~, i 6 ~} = ~.

treat

: i,

min { l~i(E) [, l~i(F) ] ) > M.

I ~i (E) I > M + sup { I ~i(S) [ : i E ~}

(EI,F I)

Then

and

{E i} E ~.

Suppose sup { l~i(E) M> 0

additive.

S

I ~i2(A2) ] > 2

and

or

sup { I

El

or FI

A1 = S \ BI.

(A2,B2)

of

B1

Now and

sup { ] ~i (H N B2) :

Proceeding by induction produces a subsequence

and a d i s j o i n t sequence

{Aj} such that

] ~ij(Aj) [ > j .

This e s t a b l i s h e s the s u f f i c i e n c y ; the necessity is c l e a r .

28 The Nikodym Boundedness Theorem f o r f i n i t e l y tions

additive

set func-

i s g i v e n by

Theorem 8.

Let

E

be a o - a l g e b r a

and f i n i t e l y additive.

If

gi

: E * R

be bounded

{gi(E)) is bounded for each E 6 K,

then (vilE) : i 6 ~, E 6 K)

Proof:

and l e t

is bound~d.

By Lemma 7, i t suffices to show that

{gi(Ej) : i, j £ N} is

bounded for each disjoint sequence {Ej} E Z.

Let

{Ej} be a dis-

joint sequence from K. Let

S(Z) be the space of

sup-norm. Each gi

E-simple functions equipped with the

induces a continuous linear functional

fi

on

I

,

S(H) via integration, = Cdgi" Consider the sequence {CEj in

S ( E ) , where CE denotes the characteristic function of

Drewnowski's Lemma I . I , any subsequence of

E.

By

{Ej} has a subsequence

{Fj} such that each {gi } is countably additive on the o-algebra g e n e r a t e d by

(Fj}.

By Theorem 2 ( i ) ,

That i s , {fi(CEj))

(CEj)

is

= (~i(Ej)}

{ f i 1- ~ c o n v e r g e n t in

S(Z).

i s bounded, and t h e r e s u l t

is established. Theorem 8 i s a l s o v a l i d f o r m e a s u r e s w i t h v a l u e s in l o c a l l y convex spaces,

and t h e v e c t o r v e r s i o n o f Theorem 8 can e a s i l y

o b t a i n e d from Theorem 8

by e m p l o y i n g t h e c l a s s i c a l

UBP ( [ 3 4 ]

be 1.3.1).

The Nikodym Boundedness Theorem i s a l s o v a l i d f o r m e a s u r e s d e f i n e d on c e r t a i n

algebras

t h a t a r e not n e c e s s a r i l y

o-algebras.

For

e x a m p l e , s e e [67] and t h e r e m a r k s f o l l o w i n g Theorem 5 . 1 0 . In c o n c l u s i o n ,

i t s h o u l d a l s o be n o t e d t h a t C o r o l l a r y

4 cannot

be used i n t h e p r o o f o f Theorem 8 g i v e n above s i n c e t h e s e q u e n c e (CEj}

is clearly

not

{] 11- X c o n v e r g e n t .

5.

Convergence of Operators

In t h i s convergent

section

we c o n s i d e r

sequences of operators

B a n a c h and S t e i n h a u s . referred

two c l a s s i c a l

We f i r s t

which are usually note that

to as the Banach-Steinhaus

the results

on t h e UBP i n s e c t i o n

Theorem 1.

(Banach-Steinhaus)

{T i } ~ L ( X , Y ) . is continuous

Proof: Let

concerning

attributed

to

one o f t h e r e s u l t s ,

Theorem, f o l l o w s

usually

i m m e d i a t e l y from

4.

Let

l i m T i x = Tx i

X

exists

be an F - s p a c e and l e t for each

x,

then

T : X ~ Y

and l i n e a r .

From C o r o l l a r y

{xj}

have

If

results

4.5,

converge to

0

the sequence in

X.

Then,

{T i}

is equicontinuous.

from t h e e q u i c o n t i n u i t y ,

l i m Tx;J = l i m l i m T i x ;J = l i m l i m T i x ;J = 0.

j

j

i

i

Hence,

T

we

is

j

continuous. We n e x t c o n s i d e r B a n a c h and S t e i n h a u s vergent

a version

Theorem i n s t e a d yields

result

39.5).

which is sometimes attributed This result

and i s u s u a l l y

also pertains presented

of the result

which is valid

under either We

any c o m p l e t e -

t h e p r o o f i s a g a i n b a s e d on t h e B a s i c M a t r i x

of the familiar

the classical

Baire category

result

To i n d i c a t e

result,

a version

to derive

Theorem and a l s o c o n s i d e r

methods.

of Banach-Steinhaus

a s an i m m e d i a t e c o r o l l a r y . we u s e i t

without

to

to con-

o r c o m p l e t e n e s s a s s u m p t i o n s on t h e domain s p a c e .

ness assumptions;

result

([47]

sequences of operators

barrelledness present

another

two c l a s s i c a l

an a p p l i c a t i o n

Our g e n e r a l for F-spaces of-~ur

general

o f t h e Nikodym C o n v e r g e n c e results

in s u m m a b i l i t y due t o

Hahn and S c h u r . We b e g i n by g i v i n g

a statement

of the classical

result

for

30 F-spaces.

The r e s u l t

i s sometimes g i v e n f o r b a r r e l l e d

s p a c e s as w e l l

(t47] 39.5).

Theorem 2. c L(X,Y)

(Banach-Steinhaus) is such t h a t

Let

X

lim Tix = Tx

-

be an F - s p a c e .

exists

If

for each

{T i }

x 6 X,

then

i

T

i s c o n t i n u o u s and

lim Tix = Tx i

b e l o n g i n g t o any compact s u b s e t o f

Again i t pleteness

is well-known t h a t

in

result

is false

For example,

i f t h e com-

the sequence

{i2ei }

m

~

= (coo)

converges pointwise to

g e n c e i s not u n i f o r m on t h e r e l a t i v e l y coo.

x

X.

this

assumption is dropped.

I

converges uniformly for

Using the n o t i o n of

Theorem 2 w h i c h i s v a l i d contains

~

0

on

coo

but t h e c o n v e r -

compact s e q u e n c e

{(llj)ej}

c o n v e r g e n c e , we f o r m u l a t e a v e r s i o n o f

i n t h e a b s e n c e o f c o m p l e t e n e s s and w h i c h

Theorem 2 a s an immediate c o r o l l a r y .

We w i l l

refer

to

Theorem 3 below as t h e G e n e r a l B a n a c h - S t e i n h a u s Theorem.

Theorem 3.

( G e n e r a l B a n a c h - S t e i n h a u s Theorem)

lim Tix = Tx i for (T

x

exists

for each

b e l o n g i n g t o any

x 6 X,

{Ti}- K

i s not assumed t o b e l o n g t o

Proof: [Tixj].

Let

be

{xj},

lim Tix j = Txj

lim Tix = Tx i {xj}

If

uniformly in

X.

L(X,Y).)

(Ti}- ~

this is a

T i 6 L(X,Y).

convergent sequence

convergent.

By the pointwise convergence of

vergence of implies

{xj}

then

Let

~

matrix.

uniformly in

j

Consider the matrix

(T i}

and the {Ti)- ~ con-

The Basic Matrix Theorem 2.2 and establishes the result.

i

One can also introduce the notion of

" ~ compactness" or "

relative compactness" and give a formulation of Theorem 3 which is more analogous to the classical statement in Theorem 2.

If

(E,T)

31

is a topological

group, then a subset

compact (¢ - ~ r e l a t i v e l y subsequence

{Xni)

x E Alx E E).

A

of

E

i s s a i d t o be

c o m p a c t ) i f e v e r y s e q u e n c e in

which i s

T - [

A

T -

has a

c o n v e r g e n t t o an e l e m e n t

Using the notion of

~

compactness,

the conclusion of

Theorem 3 can be s h a r p e n e d t o s t a t e : lim Tix = Tx

uniformly for

compact s u b s e t s o f

x

belonging to

T({Ti}) - K

relatively

X .

U s i n g Theorem 3 we can e s t a b l i s h

Theorem 2 as an immediate

corollary.

P r o o f o f Theorem 2: pact subset of suffices some

t o c o n s i d e r t h e c a s e when t h e com-

is a sequence

{xj}.

From t h e c o m p a c t n e s s ,

t o c o n s i d e r t h e c a s e when t h e s e q u e n c e

x E X.

pleteness

X

It suffices

of

The s e q u e n c e X

{xj-x}

is

II- ~

so Theorem 3 i m p l i e s t h a t

{xj)

it

converges to

c o n v e r g e n t by t h e com-

lim T i ( x j - x) = T ( x j - x) 1

u n i f o r m l y in

j.

Since

u n i f o r m l y in

j,

and t h e p r o o f o f Theorem 2 i s c o m p l e t e .

S i n c e any s e q u e n c e is

{Ti}- ~ convergent,

tim Tix = Tx, i

{xj} c X

this

which is

means

I I - ~ c o n v e r g e n t in

each

4.

x E X,

Let then

{T i}

X

Theorem 3 a l s o has t h e f o l l o w i n g c o r o l l a r y

which i s p e r h a p s more a n a l o g o u s t o t h e s t a t e m e n t

Corollary

lim Tix j = Txj i

E L(X,Y).

lim Tix j = Txj

If

in Theorem 2.

lim Tix = Tx i

u n i f o r m l y in

j

exists

for

f o r any

{{-

1

convergent sequence

Remark S. Y

{xj}

Notice that

in

neither

X.

the scalar

or the homogeneity of the operators

ments o r p r o o f s a b o v e .

Thus, t h e r e s u l t s

Ti

multiplication

in

X

and

was used in t h e s t a t e above a r e a c t u a l l y

valid

32 for additive

maps

Ti

between two normed g r o u p s

n o t known i f t h e g e n e r a l v e r s i o n s g r o u p s h a v e any u s e f u l

of the results

the utility

application

to a situation

particular,

we c o n s i d e r a c l a s s i c a l

scalar

III.7.4).

o f Nikodym, some-

and a l s o t o c e r t a i n

finitely

in s e v e r a l

additive

i s one o f t h e

and t h e o r i g i n a l

has been g e n e r a l i z e d

to vector-valued

set functions.

B r o o k s - J e w e t t Theorem ( [ 2 4 ] ,

to certain

locally

E

measures con-

t o as t h e

vector-valued

be an a l g e b r a o f s u b s e t s o f a s e t

convex m e t r i c

: ~ ~ Y

linear

if

A sequence

{~i}

lim ~ ( E j )

= 0

of additive

strongly

additive

disjoint

s e q u e n c e {Ej}.

o-algebra

space.

i s s a i d t o be s t r o n g l y

exhaustive)

if

is clearly

A finitely

additive

(strongly

set functions = 0

and l e t

additive

for each disjoint

lim ~ i ( E j ) J

finitely

been g e n e r a l i z e d ([24],

Theorem 6.

additive.

to strongly

[ 3 6 1 ) ; we f i r s t

Let

E

be a

set function

bounded o r

sequence

u n i f o r m l y in

additive,

Y

{Ej} c E.

i s s a i d t o be u n i f o r m l y

A countably additive

strongly

S

i

for each

v e c t o r measure on a

and a u n i f o r m l y c o u n t a b l y

s e q u e n c e o f v e c t o r m e a s u r e s d e f i n e d on a o - a l g e b r a

uniformly strongly

Jewett

For

set functions.

Let

additive

for

We f i r s t

sometimes r e f e r r e d

additive

result

directions.

s i d e r one o f t h e s e g e n e r a l i z a t i o n s , [36]),

In

o f Nikodym on c o n v e r g e n t

This result

in m e a s u r e t h e o r y ,

the result

is

o f Theorem 3 we g i v e an

result

m e a s u r e s has been g e n e r a l i z e d

example,

It

a b o v e f o r normed

t o as t h e Nikodym C o n v e r g e n c e Theorem,

results

Y.

where c o m p l e t e n e s s i s not p r e s e n t .

sequences of measures ([38]

most u s e f u l

and

applications.

In o r d e r t o i l l u s t r a t e

times referred

X

is also

The Nikodym C o n v e r g e n c e Theorem h a s additive

set functions

consider this

be a o - a l g e b r a .

Let

~i

by B r o o k s and

generalization.

: E ÷ Y

be s t r o n g l y

33 additive.

(i)

If

~

(ii)

lim gi(E) = g ( E ) exists for each i

is strongly

{gi }

Proof:

Let

additive

is uniformly

Z(Z)

Since each

tinuous

operator

(Note t h a t elaborate tions ~.

strongly

Y

theory required

Consider the sequence function

sequence is

{Ti}- ~

lim Ti(C E ) = lim i j i lim Ti(CEj) J uniformly For lim ~(Ej) j

i.

(i),

Let {CEj)

of

gi(Ej)

= g(Ej) = 0

= lim lim ~i(Ej) j i

~i

induces a con-

Ti~ = Js~d~i.

by i n t e g r a t i o n ,

{Ej) be a d i s j o i n t in

i s no

here since only simple func-

Z(Z),

where

sequence from CE

denotes the this

By Theorem 2, uniformly

for each

This establishes note that

each

E. By D r e w n o w s k i ' s Lemma 1 . 1 ,

convergent.

= lim ~i(Ej) 3

in

i s bounded,

equipped with

i s not assumed t o be c o m p l e t e t h e r e

are being integrated.)

characteristic

additive.

T i : S(Z) ÷ Y

even though integration

~i

then

and

be t h e s p a c e o f Z - s i m p l e f u n c t i o n s

the sup-norm. linear

E E ~,

i,

in

j.

we h a v e

Since lim ~i(Ej) J

= 0

(ii).

from the uniform convergence, = lim lim ~i(Ej) i j

= 0

so t h a t

g

is

strongly additive.

Remark 7.

Note t h a t

Theorem 6 can be e s t a b l i s h e d

B a s i c M a t r i x Theorem w i t h o u t the matrix

[gi(Ej)l

referring

directly

from the

t o Theorem 2 by c o n s i d e r i n g

and invoking the Drewnowski Lemma. The

original version of the Brooks-Jewett Theorem was established for B-spaces whereas the version above does not assume completeness and is valid for locally convex metric linear spaces (see also [36]).

We now e s t a b l i s h

the usual version

o f t h e Nikodym C o n v e r g e n c e

34 Theorem f o r v e c t o r - v a l u e d m e a s u r e s . additive Y-valued measures, countably additive

if

disjoint

{Ej).

sequence

gi : [ ~ Y'

Let

additive.

E

of countably

u n i f o r m l y in

i

for each

For t h e Nikodym Convergence Theorem, we for uniform countable additivity.

be a o - a l g e b r a .

Let

~i : E ~ Y

be c o u n t a b l y

The f o l l o w i n g a r e e q u i v a l e n t :

(i) (ii)

[gi}

is uniformly countably additive

for each

d e c r e a s i n g sequence

n Ej = ~, (iii)

Proof:

{gi )

i s s a i d t o be u n i f o r m l y

lim ~ ~ i ( E j ) = 0 n j=n

require the following criteria

Lemma 8.

A sequence

{gi)

(i)

lim g i ( E j ) J

= 0

{Ej}

from

u n i f o r m l y in

Z

with

i.

is uniformly strongly additive.

and

(ii)

are clearly

measures and ( i ) c l e a r l y Suppose ( i i i )

equivalent for countably additive

implies (iii).

h o l d s and ( i i )

fails

to hold.

Then we may assume

(by p a s s i n g t o a subsequence i f n e c e s s a r y ) t h a t t h e r e e x i s t d e c r e a s i n g sequence I gi(Fi ) I > 6 I g1(Fkl)

for each

I < 6/2.

(kj}

such t h a t

then

{Ej)

that

i.

N Fj = ~

and a

5 > 0

kI

such t h a t

There e x i s t s a

k2 > k I

such t h a t

such t h a t

C o n t i n u i n g by i n d u c t i o n p r o d u c e s a s u b s e q u e n c e

I g k j (F k j + l ) I < 6 / 2 .

is a disjoint I ~ I gkj(Fkj)

{~i }

with

Then t h e r e e x i s t s

I ~kl(Fk2 ) I < 6/2.

I gkj(Ej)

{Fj}

a

sequence from I - Ig k

+i)

If Z

Ej = Fkj \ F k j + l , with

I > 512.

But t h i s means

i s not u n i f o r m l y s t r o n g l y a d d i t i v e .

We now o b t a i n i m m e d i a t e l y t h e Nikodym Convergence Theorem f o r vector-valued

measures ([38]

III.7.4).

35 Theoz~a 9.

Let

additive.

If

(i)

Proof:

be a o - a l g e b r a .

lim ~ i ( E ) i

~

(ii)

E

= g(E)

Let

exists

is countably additive

{gi }

(ii)

gi

for each

be c o u n t a b l y

E 6 E,

then

and

is uniformly countably additive.

f o l l o w s from Theorem 6 ( i i )

For ( i ) ,

: E ÷ Y

let

{Ej}

be a d i s j o i n t

from t h e u n i f o r m c o u n t a b l e a d d i t i v i t y

and Lemma 8. s e q u e n c e from

in ( i i ) ,

~.

we h a v e

n

n

~( U E j ) = lim g i ( ~ E j ) = lim lim K g i ( E j ) j=l i j=l i n j=l so t h a t

(i)

= lim g( U E j ) n j=l

holds.

We now show t h a t

a version of the classical

Theorem (VHS) can be o b t a i n e d from t h e r e s u l t s X : ~ ~ [o,~] : Z ÷ Y, k

is a non-negative

we say t h a t

(written

such t h a t

g < < k)

[ ~(E)

we use t h e

e-6

g

(possibly

is absolutely

i f for each

[ < e

whenever

functions,

continuous with respect

t i o n a b o v e works f o r e a c h

gi"

{~i } to

X

infinite)

If set function

there e x i s t s

and

to

6 • 0

X(E) < 6. If

and

(That i s ,

{gi }

is a

is said to uniformly if the

5

in t h e d e f i n i -

U s i n g Theorem 9 we o b t a i n a v e r s i o n

o f t h e VHS Theorem f o r c o u n t a b l y a d d i t i v e ([38]

above.

continuous with respect

e > 0

E 6 ~

Vitali-Hahn-Saks

notion of absolute continuity.)

sequence of Y-valued set absolutely

Then

m e a s u r e s from Theorem 9

III.7.2).

Theorem 10. additive

~

and l e t

~i < < k E 6 ~,

Let

be a o - a l g e b r a .

k : Z )

for each

i

[o,~]

and i f

Let

~i

: K ) Y

be c o u n t a b l y

be c o u n t a b l y a d d i t i v e . lim ~ i ( E ) = ~(E) i

exists

then (i)

(ii)

~ < < k {~i }

is uniformly absolutely

continuous with

If for each

36 respect to Proof: with For

k.

F i r s t note that i f {Ej} E = n Ej

{Ej \ E}

and

is a decreasing sequence from E

X(E) = O, then

l~m gi(Ej) = 0

uniformly in

i.

is a decreasing sequence with empty intersection so

Theorem 9 and Lemma 8 imply that lim gi(Ej \ E) = lim (gi(Ej) - ~i(E)) = lim gi(Ej) = 0

J

J

uniformly in

J

i.

Now suppose that ( i i ) f a i l s to hold. such that for each F 6 E and

s u c h that

FI = ~.

sequences

6 > 0

there exist a positive integer

[ gk(F) [ ) e

and

X(F) < 5.

Put

6 > 0

k

61 = i,

and kI = 1

Then we may inductively define a subsequence { k j } and

{Fj} c E and 5j÷ 1 < 5j/2,

and

Then there is an

[ gkj+l(E) [ < e/2

(Sj} c R+ such that

[ ~kj+l(Fj+ 1) I ~ e, X(Fj÷ 1) < 5j72 whenever X(E) < 5j+ I.

~t

Ej =

~ F kk=j

Then, by the countable subadditivity,

(i)

X(Ej \ Fj) ~ XIEj+ 1) ~ so that

E X(F k) < E 5k/2 < r 5j/2 k-j+l= 5j k=j+l k=j k=j

gkj÷l(Ej+ 1 \ Fj÷ 1) < e/2.

But

[ ~kj+l(Fj÷ 1) [ =

[ ~kj÷l(Ej+ 1) - ~kj+l(Ej÷ 1 \ Fj+ I) [ > • (2)

[ gkj(Ej) [ ~ •/2.

But

{Ej} is decreasing and i f

E = fl Ej,

By the observation in the f i r s t paragraph in

i.

so that

then

X(E) = 0

lim gi(Ej) = 0 J

But t h i s contradicts (2} and ( i i ) follows.

Condition ( i ) follows immediately from ( i i ) .

by ( i ) . uniformly

37 The Nikodym C o n v e r g e n c e Theorem, t h e V i t a l i - H a h n - S a k s

Theorem

and t h e N i k o d y m B o u n d e d n e s s Theorem h a v e a l s o b e e n g e n e r a l i z e d functions

w i t h d o m a i n s w h i c h a r e not n e c e s s a r i l y

Schachermayer's treated

in d e t a i l ,

details. that

([671) t h e s e r e s u l t s

treatise

and we r e f e r

To g i v e a f u r t h e r

scalar

set

application

functions

Definition

11.

property

A family

~uasi-o-algebra

if

sequence

from

{Aj}

such that

~

U Akj fi R

This property

f o r bounded,

of subsets

o f Theorem 6 a s t h e

of a set

of sets

has a subsequence

S

property.)

is called

such that

a

each disjoint

{Ak.} J

([28]).

(for Boolean algebras)

u n d e r t h e name, s u b s e q u e n t i a l weaker form of this

finitely

o f as t h e B r o o k s - d e w e t t

i s an a l g e b r a A

are

o f o u r m a t r i x m e t h o d s we show

to the conclusion

instead

A

(and o t h e r s )

In

w h i c h a r e d e f i n e d on q u a s i - o - a l g e b r a s .

( I n [671, S c h a c h e r m a y e r r e f e r s Vitali-Hahn-Saks

o-algebras.

t h e r e a d e r t o [671 f o r more c o m p l e t e

t h e B r o o k s - d e w e t t Theorem 6 i s v a l i d

additive

to set

property

is also considered

completeness is referred

property;

to in

in [40]

a slightly

[671 a s p r o p e r t y

(E)

([67] 4.2).

Theorem 12. bounded,

Let

A

finitely

additive

l i m ~ i ( A ) = ~(A) strongly

Proof:

exists

set

function

for each

and l e t for each

A E A,

~i

: A ÷ R

i E ~.

then

{~i )

be a

If

is uniformly

additive.

First

the matrix ~.

be a q u a s i - o - a l g e b r a

assume t h a t Z = [~i(Aj)],

each where

From t h e a s s u m p t i o n t h a t

able additivity

of the

gi

{~i },

A it

is countably {Aj)

additive.

Consider

is a disjoint

sequence from

is a quasi-o-algebra

and t h e c o u n t -

follows that

Hence, f r o m t h e B a s i c M a t r i x Theorem,

Z

lim ~i(Ej) J

is a = 0

[ -matrix. uniformly for

38 i 6 ~,

and t h e r e s u l t

is established

C o n s i d e r t h e c a s e when t h e the conclusion fails, necessary) from

~

that

gi

case.

are only finitely

additive.

If

we may assume {by p a s s i n g t o a s u b s e q u e n c e i f

there exist

e > 0

and a d i s j o i n t

sequence

{Aj}

such t h a t

(3)

{ ~i(Ai)

Let

for this

Z

{ > e.

be the o-algebra generated by

~.

Extend each

bounded, finitely additive set function

~i

Lemma there is a subsequence

{Aj}

(Ak.) J

of

countably additive on the o-algebra,

on

[.

Ri

to a

By Drewnowski's

such that each

Z o, generated by the

~i

is

(Akj).

By the quasi-o-algebra assumption, we may also assume that Akj £ ~.

j=l Put

~A = (B N A : B 6 A}.

o-algebra and first

of subsets of

lim ~ i ( B ) = g(B) part,

A.

J

Each

exists

lim g i ( A k . )

a

Then

= 0

~o = ~A fl Eo

~i

is countably additive

for each

B E Ao.

u n i f o r m l y in

i.

Therefore,

(compare w i t h [67] 4 . 3 ) .

improvement o f P r o p o s i t i o n In an e n t i r e l y

fashion,

In [67]

it

i s shown t h a t

by t h e (3).

Theorem 12 a l s o g i v e s an of sets.

t h e Nikodym Boundedness Theorem

can a l s o be shown t o h o l d f o r q u a s i - o - a l g e b r a s 1.7).

~o

has t h e V i t a l i -

1.B o f [401 f o r q u a s i - o - a l g e b r a s

similar

on

This contradicts

In t h e t e r m i n o l o g y o f [671, any q u a s i - o - a l g e b r a Hahn-Saks p r o p e r t y

is a quasi-

(see

[28] Theorem

the Vitali-Hahn-Saks property always

i m p l i e s t h e Nikodym B o u n d e d n e s s P r o p e r t y ,

so we do n o t g i v e t h e p r o o f

([671 2.5). To f u r t h e r

illustrate

the applicability

of the General Banach-

S t e i n h a u s Theorem we c o n s i d e r a r e s u l t

in c l a s s i c a l

t o Schur (I50]

be an i n f i n i t e

numbers.

7.1.6)

The m a t r i x

Let A

A = [aijl

i s s a i d t o be o f c l a s s

s u m m a b i l i t y due matrix of real

(£~,c)

if

39 { j=IE a i j

xj} i

is a convergent sequence for each sequence

x = {xj} E ~ .

That i s ,

if

c

is the space of all

real

sequences

which c o n v e r g e , t h e formal m a t r i x product

Ax

c

S c h u r Theorem o f summabil-

for each sequence

x E ~.

The c l a s s i c

i t y g i v e s n e c e s s a r y and s u f f i c i e n t of class

(E~,c).

will

his proof is "classical"

and t h a t

p r o o f o f t h e t h e o r e m seems t o be known ( [ 5 0 ] ,

g i v e a p r o o f below ( o f a more g e n e r a l

Hahn [ 3 9 ] ,

analytic

A

t o be

no f u n c t i o n a l p. 168).

summability result

We

due t o

[ 6 8 ] ) b a s e d on t h e G e n e r a l B a n a c h - S t e i n h a u s Theorem;

t h e t e r m i n o l o g y o f Maddox t h i s

in

might be c o n s i d e r e d a " f u n c t i o n a l

proof".

Recall that

me

sequences with finite (me,C)

for a matrix

In h i s t r e a t m e n t o f t h e S c h u r Theorem, Maddox

makes t h e remark t h a t analytic

conditions

p r o d u c e s a s e q u e n c e in

if

matrices

Ax E c A

~

range.

a matrix

We s a y t h a t

for each

x E me.

of class

(me,C).

require

a lemma.

We f i r s t I aij

is the subspace of

i

A

of those is of class

We g i v e a c h a r a c t e r i z a t i o n

Note t h a t

c o n v e r g e s u n i f o r m l y in

consisting

a sequence of real

i f and o n l y i f f o r e a c h

of

series e > 0

J there exlsts with

{ H

N s u c h that

mino ~> N and each

jEo aij

for each finite

set

j=N

I aij I ~ 2 e

f o r each

i

([61] 1 . i . 2 . ) . ) .

Lemma 13.

Let

H

] aij

I < ~

for each

i.

If the series

j=l I aij

I

do n o t c o n v e r g e u n i f o r m l y in

i,

there exist

e > O,

J disjoint such t h a t

sequence of finite ]

H jEa i a k i J

[ > e

o

( T h i s i s immediate s i n c e i f t h e c o n d i -

i.

tion above is s a t i s f i e d , then

[ < e

sets

{o i }

for each

and a s u b s e q u e n c e i.

(k i }

40 Proof:

By t h e o b s e r v a t i o n

t aij

]

doesn't

concerning

uniform convergence above,

converge uniformly

in

i,

there

if

exists

e > 0

such

and a p o s i t i v e

integer

ki

J that

for each

with

i

there

min a i ~ i

are finite

and

] E jEa i akiJ

Applying the observation exist I

a positive

integer

E I > e jEo 1 aklJ

(4)

k1

I aij

Applying the observation

I

Jl

< e

with

I ) e.

jEo2 ak2J

a2

Note that

(4),

there

such that

that

above to the

there

exist

min a 2 > i 2 o1

a1

that

1 ( i ~ k1 .

in the paragraph

k2

] z

set

implies

such that

for

implies

set

i1 = 1

and a f i n i t e

max{max o 1 + 1, j l } = i 2 and a f i n i t e

I ~ e

above to

There exists

E J=Jl

ai E ~

and

integer

a positive

integer

such that

02

are disjoint

and by

k 2 > k 1. Induction

then produces

the sequences

in the conclusion

of the

lemma. We now e s t a b l i s h

Theorem 14. (i)

if

A E {mo,C)

K I ai

j

(ii)

the summability

lim aij

J

I

result

and only i f

converge uniformly = aj

o f Hahn.

exists

for each

in

i j.

1

Proof:

We prove f i r s t that conditions (i) and ( i i ) imply that

A E (E~,c)

so t h a t

w h e r e we may a s s u m e

in particular, Ixjl

~ 1

A E (mo,c).

for each

j.

Let

Let

{xj} E ~

e > 0.

By ( i ) ,

m

there

exists

N

such that

m ~ n ) N

implies

] j =En a i j x j

I < e

,

41 m

T h u s , by ( i i ) , verges.

(5)

I Z ajx; [ ( e j=n J

so that

the series

Z a.x. j=l J J

con-

We now h a v e ,

I Z a i,x. - E a.x. I ~ r j=l J J j=l J J j=l

I a~-~.j a~j I + I E (a i . j=N J

a~)xjl.t

N-I

(}-

I aij

j=l Since the first that

e

for

- aj

t e r m on t h e r i g h t large,

i

this

I + 2e

hand s i d e o f (5) c a n be made l e s s

shows t h a t

lim Z ai.x. = Z ajxj , i.e., A E (Z~,c) . i j=l a a j=l If

A E (m,,c),

Suppose that

then

(ii)

A E (mo,C)

holds by taking

and that (i) fails.

be as in Lemma 13 and consider the sequence {aij}j= 1

of

A

sequence m o.

{Coj}

Let the notation in

mo.

induces a continuous linear functional

Ri

Rix = j=l~ aij xj,

where

x = {xj}, and

{Coj}

x = ej E m o-

limi Rix = Rx

Each row on

exists.

is £1_ X convergent in the duality between

mo The

~1

and

j.

This

Therefore, by the General Banach-Steinhaus Theorem 3,

lim Z i Ri(Caj) " = lim i kEoj a i k = R(Coj ) = kEojZ a k implies

lim

j

Z aik = 0 l~oj

uniformly

in

i

uniformly

in

and c o n t r a d i c t s

the con-

clusion in Lemma 13. Theorem 14 hag as of Schur, i.e.,

an

immediate corollary the summability result

A E (£¢,c)

if and only if (i) and (ii) hold.

record these equivalences for later reference ([68]).

Corollary

15.

The f o l l o w i n g

(a)

AE

(~,c)

(b)

A fi (mo,c)

are equivalent:

We

by

42 (c)

(i)

and ( i i )

A consequence result

of the Hahn-Schur summability

due t o S c h u r w h i c h s t a t e s

vergent p.

o f T h e o r e m 14 h o l d .

if

170).

result:

and o n l y

if

it

converges

a sequence

in

~1

respect

by t h e s u b s p a c e

of

mo

To s e e t h a t be a s e q u e n c e sequence

in

£1.

Cauchy,

the matrix

Let

(mo,C).

to

exists

N

follows

~1

i s norm c o n of

there

for this

exists

1 ( j ( N - 1.

In Chapter for group-valued

summability

o(il,mo)

iff

the sequence

~i

E ]aij] < • j=N P P > N, JE=N l a j t

for

shows t h a t

xi ~ x

if

it

is a

induced

x i = (all}i= 1

{x i }

(i)

if

laij in

]aij

is

i ) M,

and ( i i )

of Theorem

We c l a i m By ( i ) ,

hence,

belongs

= A

to

< e I N

that

x

there

Then f r o m ( i i ) JE=N i a j ]

~ •.

~1.

Now, f r o m

for

i ) M

and

we h a v e

- ajl + j=NZ ]a i j l ÷ j=NZ lajl < 3 e, il-norm. to

Of c o u r s e ,

consider

any norm c o n v e r g e n t

o(zl,mo).

and show t h a t

above (Corollary 8, we a l s o

[aii]

o(~l,mo) -

i E ~.

and,

- aj)

8, T h e o r e m 1, we e s t a b l i s h sequences

all

~ e

is a Cauchy

the matrix

e > 0.

x = {aj}

such that

with respect

results

([SO]

of this

o(~l,mo)

let

(x i}

conditions

Let

Hence,

the result In Chapter

result,

nx i - xa I ÷ 0.

M

sequence converges

if

this

the sequence

satisfies

j~l laij - ajl ~ j=l ~ therefore,

and o n l y

be t h e s e q u e n c e g i v e n by ( i i ) .

and

that

i n norm i f

t o t h e weak t o p o l o g y

Note that

A

such that

In particular,

yields

~1

~

Thus,

x = {aj}

belongs

and,

converges

i n t h e weak t o p o l o g y to

(ii),

in

i n t h e weak t o p o l o g y

T h e o r e m 14 y i e l d s

belongs

it

a sequence

is another

T h e o r e m 14 c a n be u s e d t o g i v e an i m p r o v e d v e r s i o n

Cauchy sequence with

14.

that

result

a version the abstract

of this

result

group version

8.2). and e s t a b l i s h

o f S c h u r and Hahn f o r m a t r i c e s

versions

of the

whose e n t r i e s

are

43 elements the

of a Banach space

summability

multiplier

result

series

We c o n c l u d e result

Generalized

of Schur

in an this

in summability

section

if

A

if is

lim x i = lim Z ai.x. i i j=l J j necessary

7.1.3);

the

Theorem 16.

version

9.SAror

conditions

of

bounded

for

preserving,

a matrix are

A = {aij]

is

from the

A = [aijl A

said is

said

of Toeplitz

to be regular sufficient

regular

is

i.e.,

The r e s u l t

A also

classical

x E c , and

x = {x i } E c .

conditions

The matrix

every

another be derived

A matrix

for

convergence

that

can also

Theorem. Ax E c

for

necessary

vector

in Corollary

by indicating

due to Toeplitz

(c,c)

to be regular

is given

Another

F-space.

Banach-Steinhaus

to be of class

gives

(Theorem 8.5).

iff

([50}

(Silverman).

the

following

conditions hold: (i) (ii)

sup Z [aij [ < ® i j=l lim aij

= 0

for

each

j E

1

(iii)

Proof:

lim Z a i . = 1 i j=l J

The s u f f i c i e n c y of ( i ) - ( i i i ) ,

due to Siverman, are class-

i c a l , and we do not repeat the proof (see [50] 7.1.3). The necessity o f ( i i ) x = ej

and

and ( i i i )

x = (1,1 . . . . ).

are e a s i l y obtained by p u t t i n g

We show how the necessity of ( i ) can

be established by u t i l i z i n g the Generalized Banach-Steinhaus Theorem. Suppose that ( i ) does not hold. of finite

subsets

of

(6)

Now e a c h row

~

Then there e x i s t s a sequence, ( o i } ,

such that

sup I z i JEoi aij {aij}j= 1

of

A

max o i <

min o i + 1

and

I =

induces

a continuous

linear

functional

44 Ri

on

c

by

Rix =

exists for each

~ a i.x., j=l J J

x E c .

in the duality between Theorem

where

The sequence

~1 and

c

x = {xj} E c, and {Coj }

is

61 - X

lim Rix = Rx i convergent

so by the General Banach-Steinhaus

3, lim Ri(Caj) = lim E = R(Coj) = ~ a k i " i kEo~ ik " kEaj

uniformly in

j.

Since

and this contradicts

(6).

lim U a j kEoj ik

= 0 , we h a v e

lim U ak = 0 i kEo i

,

6.

Bilinear

Maps and H y p o c o n t i n u i t y

In t h i s

section

we c o n s i d e r

M a t r i x Theorem t o d e r i v e the joint present

continuity

Our i n i t i a l

a classical

of separately

some r e s u l t s

maps.

bilinear

pertaining results

result

in this

section

Let

E,F

and

G

biadditive

map ( i . e . ,

= b(x,y),

and

each

x

and

do n o t u s e t h e v e c t o r

the functions

b(x,.)

: E ÷ G, b ( . , y ) ( x )

of bilinear

space structure

and j o i n t

continuity

for biadditive

some r e s u l t s

are additive

vector

spaces,

which lies

for bilinear

for

on t h e h y p o c o n t i n u i t y of

F,

b

= 0

condition:

uniformly for

0

V in

of

if

be t h e w e a k e s t t o p o l o g y on

map

b(.,y),

y E F, and

b,

is continuous. but this

s h o u l d c a u s e no d i f f i c u l t i e s . ) family of all

o(E,F)-

E

G

and e a c h

in

this E

is

and

such that

N E R,

each addi-

(The t o p o l o g y a ( E , F )

is suppressed

o(F,E)

~ convergent

in

y E N.

o(E,F)

G

groups and

such that

xi ÷ 0

Let

d e p e n d s upon

E

0

Since the groups involved are metrizable,

lim b(xi,Y)

of

an

i s s a i d t o be

b(U,N) c V.

to the following

We d e f i n e

o f s u c h maps.

for each neighborhood U

Bourbaki

between sepa-

maps.

is a neighborhood

tive

be a

: F ÷ G, b ( x , . ) ( y )

N E R, t h e r e

then

pre-

b : E x F ÷ G

maps b e t w e e n t o p o l o g i c a l

is a family of subsets

K-hypocontinuous if

equivalent

of

Many o f t h e

but the results

= b(x,y),

maps b e t w e e n t o p o l o g i c a l

analogous notion

R

We a l s o

more g e n e r a l .

the notion of hypocontinuity

continuity

If

in [75],

on

y).

has introduced

establish

maps.

f o r normed g r o u p s .

be normed g r o u p s and l e t

b(-,y)

For bilinear

rate

bilinear

to the hypocontinuity

are contained

sented here are slightly

o f Mazur and O r l i c z

continuous

t h e s p a c e s i n v o l v e d and so a r e v a l i d results

maps and u s e t h e B a s i c

in the notation

is defined

sequences in

also

similarly. E

and

The

i s d e n o t e d by

46 X(E,F) (K(F,E)

is defined similarly).

We now s t a t e

our first

o f Theorem 5 . 3 f o r t h i s

Theorem 1.

Let

and { y j }

is

uniformly

Proof:

b

a restatement

continuous. then

If

xi ~ 0

in

o(E,F)

lim b(xi,Y J) = 0 i

j.

continuous,

convergent,

is essentially

setting.

X convergent,

Consider the matrix

separately

it

particular

be s e p a r a t e l y

o(F,E)-

in

result;

it

[b(xi,Yj)].

xi * 0

follows

that

in

this

From t h e f a c t s

o(E,F)

matrix

and

is a

K

{yj}

that is

matrix.

b

is

o(F,E)-

K

The B a s i c

M a t r i x Theorem 2 . 2 g i v e s t h e r e s u l t . We now p r e s e n t

several

hypocontinuity-type

corollaries

of

Theorem 1.

Corollary

2.

Let

b

be s e p a r a t e l y

continuous.

Then

b

is

~(F,E)-hypocontinuous.

Proof:

This follows

in

then

E, Let

xi ÷ 0

X(F)

(with respect topology of

Corollary

in

to the original is stronger

Let

b

if

xi * 0

o(E,F).

be t h e f a m i l y o f a l l

F

3.

i m m e d i a t e l y f r o m Theorem 1 s i n c e

~

topology of than

o(F,E),

be s e p a r a t e l y

convergent sequences F).

F

Since the metric

K(F) E K ( F , E ) ,

continuous.

in

Then

b

and we h a v e

is

K(F)-hypocontinuous.

Corollary topologies

on

generalization

3, w h i c h i s a r e s u l t E

and

F,

concerning only the original

now y i e l d s

of a classical

result

the following on j o i n t

very interesting

continuity.

47 Corollary

4.

plete.

Then

Proof:

Let

Let b

b

be s e p a r a t e l y

is continuous

x i ÷ 0 (Yi ÷ 0)

continuous

(i.e.,

in

jointly

E (F).

is

K convergent.

By Corollary 3,

j.

In p a r t i c u l a r ,

lim b(xi,Yi ) = 0, i

and l e t

F

b e com-

continuous).

Since

F i s complete,

lim b(xi,Yj ) = 0 i i.e.,

b

(yi }

uniformly in

i s continuous.

For the case of metric l i n e a r spaces, Corollary 4 seems to be o r i g i n a l l y due to Mazur and Orlicz ([51]).

The r e s u l t seems to have

been rediscovered many times and i s sometimes a t t r i b u t e d to Bourbaki ([22] 15.14) although the Bourbaki version requires the completeness of both of the (metric l i n e a r ) spaces

E

and

F.

Baire category

methods are usually u t i l i z e d in the proofs of Corollary 4 found in the l i t e r a t u r e (cf I22] 15.14) as contrasted with the matrix methods used above.

In the absence of completeness there do not appear to be

any hypocontinuous-type of r e s u l t s of the nature of C o r o l l a r i e s 2 and 3.

Of course, i t should also be noted that the r e s u l t s above are

v a l i d for normed groups whereas the c l a s s i c a l r e s u l t s pertaining to b i l i n e a r maps in the l i t e r a t u r e consider the case of (complete) metric l i n e a r spaces. As an i n t e r e s t i n g a p p l i c a t i o n of Corollary 4, we consider the d e f i n i t i o n of a metrizable l i n e a r space due to Banach ([20], [46] 15.13).

Banach developes a theory of metrizable l i n e a r spaces under

the following set of axioms which (formally) appear to be weaker than those for a metric l i n e a r space. t r a n s l a t i o n invariant metric

d

Let

X be a vector space with a

defined on

X which s a t i s f i e s the

following p r o p e r t i e s : (a)

tnX * 0

for

each

(b)

tx n * 0

for

each scalar

(c)

X

is complete with

metric

d.

x E X

respect

if t

tn * 0 if

xn * 0

to the translation

invariant

48 I t c a n be shown t h a t i n d u c e d by

d

the vector space

is actually

a topological

axioms ( a ) - ( c ) a r e s a t i s f i e d . immediately that is a toplogical necessary.

if

X

under the topology

vector space ([46]

Using C o r o l l a r y

satisfies

vector space,

X

4,

it

The p r o o f o f t h e f a c t

then

t h e c o m p l e t e n e s s in ( c ) that

X

if

follows

o n l y axioms (a) and ( b ) ,

i.e.,

15.13)

is a topological

X

i s not vector

s p a c e g i v e n i n [46] u s e s t h e B a i r e C a t e g o r y Theorem and, t h e r e f o r e , will

not y i e l d

the result

A somewhat s i m i l a r carried X

o u t i n [80]

i n t h e a b s e n c e o f axiom ( c ) .

discussion

1.2.

If

is a non-negative function (1)

II x II ) 0

(2~

11 x + y

(3~

II

-

and

concerning quasi-normed spaces is

X

is a vector space,

II

III on

II x Ill = 0

I1

=

II

and (a) and ( b ) .

It

i s shown in [80]

space is a topological on r e s u l t s

x

satisfying:

iff

x : 0

11

I1 ~ II x I1 + II y

x

relies

X

a q u a s i - n o r m on

II

1.2.2,

vector space.

that

any q u a s i - n o r m e d

The method o f p r o o f in [80]

o f measure t h e o r y and s h o u l d be c o n t r a s t e d

with the

e l e m e n t a r y m a t r i x methods employed a b o v e . We now c o n s i d e r t h e c a s e when t h e s p a c e s ological this

vector

case,

family of

i.e.,

K

bounded s e t s .

We l e t

(similarly for

K~(F,E)),

bounded subsets of

E

a(E,F)-

and let

following analogue of Corollary 2

Corollary

5.

Let

b

be s e p a r a t e l y

X~(F,E)-hypocontinuous.

notation.

for

be a b i l i n e a r Further,

let E

We then have the

bounded sets.

continuous.

In

for the

be the family of all

~(F)). ~

are top-

spaces.

K bounded s u b s e t s o f

~(E)

(similarly for

G

2 and 3 a r e v a l i d

b : E x F ÷ G

the previous topological

be t h e f a m i l y o f a l l

and

in o u r c a s e m e t r i c l i n e a r

the analogues of Corollaries

map and r e t a i n K~(E,F)

spaces,

E, F

Then

b

is

X

49 Proof:

Let

xj * 0

u n i f o r m l y in

j

in

E.

for each

It suffices o(F,E)- X

If this condition fails bounded sequence exist

(yj}

k i > i, Ji

i I = 1,

there exist

lim b(xi,Y j) = 0 i Ib(xi,Yj) I < e k 2 • i 2 and

J2

such t h a t

Jl

j,

i ) i2

[b(Xkl, y j l ) l i2

i if

• e.

Now s i n c e

i2

Thus

there exist J2 • 31"

can be c o n t i n u e d t o produce subsequences

bounded sequence

( y i }.

P i c k a sequence o f p o s i t i v e Then

lary 2 implies that

Thus, i t s u f f i c e s xi * 0

scalars

( ( 1 / t i ) Y i}

{t i}

E

(k i )

and

t o show t h a t

and e a c h

such t h a t

o(F,E)-~

ti * =

o ( F , E ) - ~ convergent so C o r o l -

is

lim b ( t i x i ,

in

there

such t h a t

For

Ib(Xk2, y j 2 ) l • e.

f o r e a c h sequence

the result

o(F,E)-.~

In p a r t i c u l a r ,

1 ~ j ( Jl"

Ib(Xki, Y J i ) I • e.

t i x i * 0.

> e.

with

lim b ( x i , Y i ) = 0

and

a

{yj} ~ F.

such t h a t f o r e a c h

there exists

and

such t h a t

This c o n s t r u c t i o n {ji )

and

for each for

e > 0

Ib(Xki, y j i ) [

kI

lim b ( x i , Y j ) = 0 i

bounded sequence

to hold, there exist

and an

with

t o show t h a t

( i / t i ) Y i ) = lim b ( x i , Y i) = 0,

and

is established.

Since

~(F)

for the original

C o r o l l a r y 6.

~ ~(F,E),

C o r o l l a r y 5 has t h e f o l l o w i n g c o r o l l a r y

topologies of

Let

b

E

and

F.

be s e p a r a t e l y c o n t i n u o u s .

Then

b

is

~(F)-hypocontinuous.

In p a r t i c u l a r , is

if

F

~(F)-hypocontinuous

subsets o f

F

([22],

is complete, Corollary 5 implies that when

~(F)

b

i s t h e f a m i l y o f a l l bounded

[47]).

Note t h a t t h e r e a r e no completeness or b a r r e l l e d n e s s assumptions

in e i t h e r C o r o l l a r y fi o r 6.

The t y p i c a l r e s u l t

in t h e l i t e r a t u r e

50 asserting

some t y p e o f h y p o c o n t i n u i t y has some s o r t o f c o m p l e t e n e s s

or b a r r e l l e d n e s s

a s s u m p t i o n on t h e s p a c e s i n v o l v e d ( f o r example,

[47]

40.2). We n e x t c o n s i d e r some r e s u l t s on t h e t o p o l o g i c a l eralization bilinear

of a result

maps ( [ 2 2 ]

Theorem 7.

Let

e a r maps from for each

{b i )

into

G

such that

Then

(b i}

A (B)

is the range of a

(ii)

A (B)

is a

and

Fix

F,

K

Let

of

continuous bilin: i)

i s bounded

i s u n i f o r m l y bounded on e a c h

R

{xi},

and c o n s i d e r operators

u n i f o r m l y bounded on (bi(xj,y)

By C o r o l l a r y

4.4,

sequences in

F.

~

: i,j)

family of operators

convergent sequence in

{yi }

E (F).

E (F).

be

K

convergent sequences

bi(.,y)

This sequence of

i s p o i n t w i s e bounded and,

i s bounded f o r e a c h .)

: i,j)

family is

Thus,

: E * C.

c o n v e r g e n t s e q u e n c e s in

{bi(x j,

this

E (Corollary 4.4).

y E F,

i.e.,

the

i s p o i n t w i s e bounded on

u n i f o r m l y bounded on

( b i ( x j , y k) : i ,

therefore,

j,

k}

K

F.

convergent

i s bounded.

This

(i).

Condition (ii) Corollary

give a gen-

respectively.

y E F

establishes

{bi(x,y)

bounded s u b s e t o f

(i).

continuous linear

Hence,

Our r e s u l t s

when

(i)

E

F.

be a s e q u e n c e o f s e p a r a t e l y

E x F

Consider

and

15.14.3).

A x B c E x F

Proof:

E

mappings

of Bourbaki f o r e q u i c o n t i n u o u s f a m i l i e s

( x , y ) E E x F.

product

in

vector spaces

for sequences of bilinear

is established

in a s i m i l a r

fashion using

4.4 (ii).

Theorem 7 c a n be viewed as a v e r s i o n o f t h e G e n e r a l i z e d U n i f o r m Boundedness P r i n c i p l e tinuous bilinear

maps.

(Theorem 4 . 2 ) (In this

for sequences of separately

regard a l s o see C o r o l l a r y

F o r c o m p l e t e s p a c e s we have t h e f o l l o w i n g

con-

15 b e l o w . )

51 Corollary

8.

Let

complete,

then E

{b i }

(b i}

subsets

of

and

Proof:

The c o r o l l a r y

be a s in Theorem 7.

follows

result

is

J~ bounded i f f

it

8 with 40.4.9

If

a r e normed s p a c e s ,

and

G

the sequence

{b i }

of [47].

where

x = (xi),

E

is uni-

F,

and,

8 gives the Bourbaki

the completeness

assump-

i =jE=lxjy 3

y = ( y i }.

Then e a c h

bi

is separately

(b i }

is pointwise

sequence with a where

fi

Let

i

i = E ej, j=l

(e} x ( f i

bounded on

so t h a t

which c o n v e r g e s to

0

x Coo.

If

e E ~=

then

the sequence

(b i}

i s not

which gives the general

in

form

15.14.3.

be a s i n Theorem 7. uniformly

continuous,

: i E ~{}.

g i v e n i n [22]

(b i }

~

in e a c h c o o r d i n a t e ,

another corollary

lim bi(xj,y j) = 0 J

convergent.

it

and

bi(x,y)

of the Bourbaki result

then

E

by

We n e x t p r e s e n t

I0.

in

x Coo ~ R

bounded on t h e p r o d u c t

Corollary

Corollary

since

bi : ~

and t h e s e q u e n c e

bi(e,f i) = i,

bails

in Corollary

8 c a n n o t be c o m p l e t e l y d r o p p e d .

Define

is the constant

the conclusion

an e x a m p l e which shows t h a t

in C o r o l l a r y

Example 9.

in

15.14.3.

We p r e s e n t tion

are

i s bounded.

is equicontinuous

in t h e c a s e o f normed s p a c e s , in [22]

F

i m m e d i a t e l y f r o m Theorem 7 s i n c e

f o r m l y bounded on t h e p r o d u c t o f t h e u n i t thus,

and

F.

Compare C o r o l l a r y

8 implies that

E

i s u n i f o r m l y bounded on p r o d u c t s o f bounded

complete spaces a set

E, F

If

i

If

E

is complete,

for each sequence

and e a c h s e q u e n c e { y j )

in

F

(xj)

which is

in

52 Proof: in

If the conclusion fails,

G

with

such that

for each

i

there exists

there are positive

integers

U

mi ,

of 0

ni ) i

b m i ( X n i , Y n i ) ~ U. For

i1 = 1

By C o r o l l a r y

such t h a t Thus t h i s {mi}

there exist

3 there exists

b i ( x j , y j ) fi U

for

1 ~ i ~ m1.

construction {n i}

t i ÷ ~.

ness of

b m l ( x n l , Ynl) ~ U.

such t h a t

j ~ i2

Now t h e r e e x i s t

Note t h a t

implies

n2 > i 2

m2 > m1

and

and

m2

n 2 > n 1.

can be c o n t i n u e d t o p r o d u c e two s u b s e q u e n c e s

such that

b m i ( X n i , Y n i ) ~ U.

The s e q u e n c e

scalars

{tiXni}

E so by Theorem 7,

(1/ti)bmi(tiXni,Yni

such that

i 2 > n1

bm2(Xn2,Yn2) ~ U.

and

m1, n 1

Pick a sequence of positive and

a neighborhood

is

{t i} K

tiXni * 0

c o n v e r g e n t by t h e c o m p l e t e -

{bmi(tiXni,Yni)}

} = bmi(Xni,Yni) ÷ 0

such that

i s bounded.

contradicting

Thus,

the construc-

tion above. Corollary its

full

generality

Corolla~-y I1. complete,

Proof: {yi} in

10 now y i e l d s

Let is

i,

K i.e.,

{b i }

be a s in Theorem 7.

xi * 0

in

E

convergent. {b i}

and

Yi * 0

By C o r o l l a r y

in

If

E

and

F

are

in

F.

By t h e c o m p l e t e n e s s ,

10, lim b i ( x j , y j ) = 0 J

uniformly

is equicontinuous. in Example 9 a g a i n shows t h a t t h e com-

a s s u m p t i o n in C o r o l l a r y

(The c a l c u l a t i o n s {b i}

15.14.3)

is equicontinuous.

The example p r e s e n t e d pleteness

([22]

as an i m m e d i a t e c o r o l l a r y .

Let

{b i}

t h e Bourbaki r e s u l t

11 c a n n o t be c o m p l e t e l y d r o p p e d .

g i v e n below in Example 13 show t h a t t h e s e q u e n c e

i s not e q u i c o n t i n u o u s . )

53 Corollary

I0 also yields

Theorem f o r b i l i n e a r

Corollary

12.

lim bi(x,y) i

Let

maps ( s e e 4 . 5 and 5 . 1 ) .

{b i }

= b(x,y)

are complete,

then

be s e p a r a t e l y

exists {b i}

The e q u i c o n t i n u i t y

E

yj * 0

ness of

F

in

F,

so Corollary

for each

follows

then

j

b

{yj)

is

10 i m p l i e s

K

If

E

and

F

is continuous.

I1.

If

xj * 0

in

c o n v e r g e n t by t h e c o m p l e t e -

that and

b

is continuous.

i the completeness

assump-

c a n n o t be c o m p l e t e l y d r o p p e d . We now c o n s i d e r

linear

maps.

E x F * G, of linear

(y E F ) .

some r e s u l t s

on s e p a r a t e

equicontinuity

If

{b i )

is a sequence of bilinear

then

(b i}

is right

maps

{bi(x,-)

family of linear

right

and

from Corollary

The s e q u e n c e in Example g shows t h a t tions

and s u c h t h a t

( x , y ) E E x F.

l i m b(x~,y;)jj = l i m l i m bi(x~,y~)jj = O,

j

continuous

is equicontinuous

Proof: and

an a n a l o g u e o f t h e B a n a c h - S t e i n h a u s

: i}

maps f r o m

F

(left) ({bi(.,y} into

The s e q u e n c e i s s e p a r a t e l y

and l e f t

fi

maps f r o m

equicontinuous : i}) (E

if the family

i s an e q u i c o n t i n u o u s

into

G)

equicontinuous

for each if

it

x E E

is both

equicontinuous.

We g i v e an e x a m p l e o f a s e q u e n c e o f b i l i n e a r equicontinuous

for bi-

but not s e p a r a t e l y

Example 13.

Let

{b i }

assume t h a t

yj = 0

maps which i s l e f t

equicontinuous.

be a s in Example 9.

Fix

y

=

{ y j } E Coo

and

n

for

j ~ n.

x = {xj} E t.=° so that

{bi(.,y)} If

for

is equicontinuous.

Then

f o r i i> n

n {{bi(-,y){{ ~< E [yj{ j=l

Hence,

{bi)

e E ~=o is the constant sequence with a

bi(x,y)

=

~" x ~ y ; j=l JJ

and, t h e r e f o r e ,

is l e f t equicontinuous. 1

in each coordinate,

54

then

]{bi(e,.)]{ = i

That i s ,

so that

{bi(e,-)}

is not equicontinuous.

{bi } i s not right equicontinuous and, therefore, not

separately equicontinuous.

We next o b s e r v e t h a t a l e f t e q u i c o n t i n u o u s sequence o f b i l i n e a r maps i s p o i n t w i s e bounded.

Proposition 14. {bi}

Let

bi : E x F ÷ G be l e f t equicontinuous.

Then

is pointwise bounded.

Proof: that

Let

(x,y) @E x F

and

t i b i ( x , y ) = bi(tix,Y) ~ O.

{t i} 6 co .

Then t i x ÷ 0

Hence, {bi(x,y)}

in

E

so

is bounded.

From Proposition 14, i t follows that the conclusions of Theorem 7 and Corollaries 8, i0 and i i hold for sequences of separately continuous, l e f t equicontinuous

b i l i n e a r maps.

Note that i t follows from Corollary 4 . 5 . ( i i ) that i f

{bi}

is a

pointwise bounded sequence of separately continuous b i l i n e a r maps and if

E

is complete, then

{bi]

is l e f t equicontinuous.

Combining

t h i s observation with Proposition 14 gives the following corollary.

C o r o l l a r y 15. plete,

Let

{b i )

be s e p a r a t e l y c o n t i n u o u s .

If

E

is com-

the following are equivalent:

(i)

{b i}

is left equicontinuous

(ii)

{b i}

i s p o i n t w i s e bounded.

The sequence in Example 13 shows that the completeness assumption on Coo

and

E

in Corollary 15 cannot be dropped. Z~

(Reverse the spaces

in Example 13).

Corollary 15 can be viewed as a version of the Uniform Boundedness Principle for sequences of separately continuous bilinear maps. (See Corollary 4.5.(ii).)

55 We now e s t a b l i s h F-spaces.

a generalized

Since a left

uous b i l i n e a r

v e r s i o n o f 4 0 . 2 . 2 o f [47] f o r

equicontinuous sequence of separately

functions

i s p o i n t w i s e bounded, t h e c o r o l l a r y

f o l l o w s i m m e d i a t e l y from C o r o l l a r y

11.

continbelow

However, we g i v e a d i r e c t

p r o o f b a s e d o n l y on t h e B a s i c M a t r i x Theorem.

Corollary

16.

Let

equicontinuous.

bi : E x F ~ G

If

E

and

F

be s e p a r a t e l y

c o n t i n u o u s and l e f t

are complete, then

{b i}

is equicon-

tinuous.

Proof:

Let

{xj}

and

{yj}

converge to

0

in

E

and

F, r e s p e c -

tively. Consider the matrix tinuity

lim z i j

= 0

[zij]

= [bi(xi,Yj)].

for each

j.

If

By t h e l e f t

{mj}

equicon-

i s any i n c r e a s i n g

i

sequence of positive

integers,

subsequence

{nj}

the separate

continuity,

equicontinuity [zij]

is a

lim z i i i

by t h e c o m p l e t e n e s s o f

such t h a t t h e s e r i e s

K

matrix.

yn.),j

lim ~ Z i n j = 0. 1 J

and t h e r e s u l t

For the case of complete metric linear

separately result

has a By

and t h e l e f t That i s ,

the matrix

By t h e B a s i c M a t r i x Theorem,

= lim b i ( x i , Y i ) = 0,

gives a generalization

{mj}

~ Ynj i s c o n v e r g e n t .

Zj Zin j = b i ( x i ,

then implies that

F

follows. spaces, Corollary

o f 4 0 . 2 . 2 o f [47] where i t

equicontinuous

i s shown t h a t any

sequence is equicontinuous.

above u s e s o n l y t h e e q u i c o n t i n u i t y

16

Note t h a t t h e

in one o f t h e v a r i a b l e s .

The s e q u e n c e in Example 13 shows t h a t t h e c o m p l e t e n e s s a s s u m p t i o n c a n n o t be c o m p l e t e l y e l i m i n a t e d . Finally,

we c o n c l u d e t h i s

uniform hypocontinuity bilinear

maps from

section

of bilinear

E x F

into

by c o n s i d e r i n g

maps. G

Let

and l e t

{b i} ~

a result

on

be a s e q u e n c e o f

be a f a m i l y o f s u b -

56 sets of

F.

The s e q u e n c e

{b i}

i f g i v e n any n e i g h b o r h o o d a neighborhood

V

We now e s t a b l i s h

of

0

left

in

Let

E

bi

Then

(ii)

~(F)-equihypocontinuous

j

when

q u e n c e , t h e n by t h e

K

such t h a t

By t h e s e p a r a t e

continuity

liml ~J Z i n j = liml b i ( x i '

matrix. in

for each

j

when

3

and

for all

S

is i.

for

and

{yj} i s

K

If

{yj}

{mj} there

~j Zin j exists

uni-

convergent

in

F.

i s any s u b s e is a subsequence

~ Ynj c o n v e r g e s t o an element

y) j

c o n t i n u o u s and

lim b i ( x i , Y j ) = 0 i

= [bi(x i, yj)].

y

of

F.

c o n v e r g e s and by t h e l e f t

by t h e l e f t

it suffices xi * 0

in

to hold, then,

a neighborhood

such t h a t {t i}

bi(V,N) E U

there

equicontinuity.

equicontinuity, lim b i { x i , i

[zij]

yj) = 0

Since is a

uniformly

j.

tion fails exist

E

By t h e B a s i c M a t r i x Theorem,

For ( i i ) , in

in

convergence of

{mj}

lim z i j = 0 1

N E K,

is

t o show t h a t

xi ~ 0 [zij]

of

and any

and

it suffices

Consider the matrix

{nj}

G

be s e p a r a t e l y

{b i}

K(F)-equihypocontinuous

f o r m l y in

in

such t h a t

E x F ÷ G

:

(i)

For ( i ) ,

0

maps.

equicontinuous.

Proof:

of

~-equihypocontiauous

the analogues of Corollaries

sequences of bilinear

Theorem 17.

U

i s s a i d t o be

U

t o show t h a t E

and

ti * ~

is

~

bounded.

as in t h e p r o o f o f C o r o l l a r y of 0

bmi(Xmi,Yni) ~ U.

such that

{yj}

lim b i ( x i , Y j ) = 0 i

and

in

G

If this

{mi}, {n i}

Pick a sequence of positive Then

condi-

10, t h e r e

and s u b s e q u e n c e s

tlXmi. * O.

uniform-

{(1/ti)Ynl}.

scalars is

K

57 c o n v e r g e n t SO by t h e f i r s t limi b m i ( t i x m i '

part, (I/ti)Yni)

= limi bm1("x m i ' Y n i ) = 0.

This gives the desired contradiction. For t h e c a s e when 17 y i e l d s 4 0 . 2 . 3 ( b ) immediate c o r o l l a r y .

i s a complete m e t r i c l i n e a r

F

o f [47]

(or Proposition

That i s ,

equihypocontinuous with respect F.

if

F

result.

is

in one o f t h e v a r i a b l e s

i s used

I t i s a l s o w o r t h w h i l e n o t i n g t h a t Theorem 17 i s

the non-locally Finally,

{b i )

t o t h e f a m i l y o f bounded s u b s e t s o f

v a l i d w i t h no c o m p l e t e n e s s o r b a r r e l l e d n e s s that

o f [ 2 2 ] ) a s an

is complete, then

Note t h a t o n l y t h e e q u i c o n t i n u i t y

in t h i s

III.4.14

s p a c e , Theorem

assumptions,

c o n v e x c a s e i s c o v e r e d in Theorem 17.

note that

if both

E

and

F

a r e c o m p l e t e and

i s a s e q u e n c e o f p o i n t w i s e bounded, s e p a r a t e l y maps, t h e n C o r o l l a r y

and a l s o

continuous bilinear

1S and Theorem 17 imply t h a t

pocontinuous with respect

{b i }

is equihy-

t o t h e f a m i l y o f bounded s u b s e t s o f

(Compare w i t h E x e r c i s e 4 0 . 8 . c o f [ 2 2 ] . )

{b i }

F.

7.

Orlicz-Pettis

In t h i s

Theorems

section

to Orlicz-Pettis

we c o n s i d e r t h e a p p l i c a t i o n

type theorems.

The c l a s s i c a l

f o r normed s p a c e s has been c o n s i d e r e d of a result Pettis

on

K

subseries

that

l y p r o v e n by O r l i c z and t h e f i r s t available

any s e r i e s

proof of the result

An O r l i c z - P e t t i s

a series

actually

subseries was o r i g i n a l -

normed s p a c e s w h i c h was

asserts

c o n v e r g e n t i n some t o p o l o g y i s topology.

type results.

The l i t e r -

We w i l l

present

w h i c h can be o b t a i n e d by m a t r i x m e t h o d s , but no

be made t o g i v e a c o m p l e t e s u r v e y o f s u c h O r l i c z - P e t t i s For h i s t o r i c a l

Orlicz-Pettis

results,

Recall that subseries

is a theorem that

c o n v e r g e n t i n some s t r o n g e r

such r e s u l t s

type results.

r e m a r k s and e x t e n s i v e

the reader

a series

Zxj

is referred

E.

in t h e t o p o l o g i c a l

When t h e r e

references

t o [34]

convergent if for each subsequence

c o n v e r g e s in E,

Orlicz-

weakly complete spaces ( [ 5 5 ] } ,

type of result

which is subseries

subseries

attempt will

on

The c l a s s i c a l

This result

for general

a t u r e abounds w i t h s u c h O r l i c z - P e t t i s

Zx,.

3 as an a p p l i c a t i o n

i n a normed s p a c e w h i c h i s

norm t o p o l o g y .

for sequentially

Theorem

t o n o n - P o l i s h s p e a k i n g m a t h e m a t i c i a n s was g i v e n by P e t t i s

([591).

several

in s e c t i o n

c o n v e r g e n t f o r t h e weak t o p o l o g y i s a c t u a l l y

convergent for the stronger

that

Orlicz-Pettis

convergent sequences {3.7).

Theorem g u a r a n t e e s

o f m a t r i x methods

1.6 or [31].

group

{Xk.}, J

to

E

is

the subseries

i s more t h a n one g r o u p t o p o l o g y

we have t h e f o l l o w i n g c r i t e r i a

for subseries

c o n v e r g e n c e in

both topologies.

Theorem 1.

Let

the original which c o n s i s t s

o

be a g r o u p t o p o l o g y on

topology of

E

and s u c h t h a t

o f s e t s which are c l o s e d f o r

E

w h i c h i s weaker t h a n

E

has a b a s i s

o.

at

If each series

0 Hx: 1

59 in

E

which is s u b s e r i e s

then each series subseries

in

E

convergent for

o

which is s u b s e r i e s

convergent for the original

result

II.1

or [30],

Orlicz-Pettis

in 3 . 8 as a c o r o l l a r y

g e n c e ; h e r e we g i v e a s i m p l e d i r e c t

o

= O,

is also

topology.

consider the classical

was d e r i v e d

lim[xi[

convergent for

For t h e p r o o f o f Theorem 1, s e e [ 7 7 ] , We f i r s t

satisfies

IV.I.1.

Theorem.

of a result

on

This

X

conver-

p r o o f b a s e d on t h e B a s i c M a t r i x

Theorem.

Theorem 2.

(Orlicz-Pettis)

Let

X

be a normed s p a c e and l e t

be s u b s e r i e s

convergent with respect

t o t h e weak t o p o l o g y .

is subseries

convergent with respect

t o t h e norm t o p o l o g y .

Proof:

By r e p l a c i n g

we may assume t h a t Let

{Xn'}l

We claim that

X X

zi

=

SPi

{x i}

and s e t

i s a norm Cauchy sequence.

sequence of positive

= Spi+l

Pi÷l Z

k=Pi+IXnk ,

integers

and n o t e t h a t I

that

[Izil I ÷

0.

Ex i

{xi},

is separable.

be an i n c r e a s i n g -

Then

by t h e c l o s e d s u b s p a c e spanned by t h e

be a s u b s e q u e n c e o f

{s i}

Zx i

For each

i,

pick

i s i = k=lZ xnk.

For t h i s ,

let

{pi )

and it

suffices

t o show

I

i

z i 6 X such that

llzil I = l

I

and

< z i , z i > = [ [ z i [ 1.

Since

X

0

Theorem,

{z i} z

by t h e B a n a c h - A l a o g l u

J

has a s u b s e q u e n c e

I

element

is separable, {Zki}

w h i c h c o n v e r g e s weak ~ t o an

I

E X . i

Consider the matrix

[zij]

= [l. J

By t h e weak ~ c o n v e r -

g

gence of

matrix that

{Zki }

[zij]

and t h e weak s u b s e r i e s

is a

~

matrix.

lim z i i : lim llZki[l = O.

convergence of

Zz i ,

the

The Basic hiatrix Theorem implies Since the same argument can be

60 applied

to each subsequence of

Since that

{s i}

Ex i

, this

The l o c a l l y

Jlzill it

* 0.

follows

convergent.

convex version

of the 0rlicz-Pettis

Theorem f o l l o w s

f r o m Theorem 2 by s i m p l y a p p l y i n g Theorem 2 t o e a c h c o n t i n -

uous s e m i - n o r m on t h e s p a c e .

That is,

s p a c e and i f t h e s e r i e s

is subseries

t h e weak t o p o l o g y o f

has established locally

Ex i

if

X, t h e n t h e s e r i e s

vergent with respect

nature

shows t h a t

i s norm Cauchy and w e a k l y c o n v e r g e n t ,

i s norm s u b s e r i e s

directly

{z i )

X

is a locally

convergent with respect is also subseries

t o t h e Mackey t o p o l o g y o f

a very general

convex spaces.

X.

to

con-

In [ 7 8 ] ,

form of the Orlicz-Pettis

We now g i v e a r e s u l t

as Tweddle's result

convex

Tweddle

topology for

o f somewhat t h e same

w h i c h c a n be d e r i v e d

from the Basic Matrix

Theorem. Let

E

be a l o c a l l y

c o n v e x s p a c e and l e t

J

be t h e f a m i l y o f I

all

series

the vector

in

E

w h i c h a r e weak s u b s e r i e s

space of all

$

linear

convergent.

functionals

y

on

Let

E

w

for all

respect

duality

Ex i E J .

Clearly

E

be

such that

I

= E i i

G

I

c

O .

With

0

to the natural

following Orlicz-Pettis

Theorem 3. vex s p a c e

If E,

E

and

fi ,

we h a v e t h e

type result.

Zx i then

between

i s weak s u b s e r i e s

convergent

Ex i

convergent with respect

is subseries

in t h e l o c a l l y

conto the

0

t o p o l o g y o f u n i f o r m c o n v e r g e n c e on

Proof:

Let

{yj} ~ G

be

I

o(G , E ) - C a u c h y s e q u e n c e s i n

o(G , E ) - C a u c h y .

Consider the matrix

J

[zij]

I

= [l.

Now

lim zij

exists

since

s

(yj}

is

o(G ,E)-

1

Cauchy.

F o r any s u b s e q u e n c e

{kj),

{. }

converges since

[zij]

matrix.

is a

K

the sequence

{yi }

is

G .

{Zj Z i k j} =

olG ,E)-Cauchy.

Thus,

The B a s i c M a t r i x Theorem i m p l i e s t h a t

61

lim z i j = 0 J

uniformly in

uniform convergence on

i, i . e . ,

lim xj = 0 J

in the topology of

a(G ,E)-Cauchy sequences.

Theorem 1 now

gives the r e s u l t . Tweddle's O r l i c z - P e t t i s r e s u l t i s somewhat d i f f e r e n t than Theorem 3 and in some sense i s the strongest type of O r l i c z - P e t t i s r e s u l t that can be obtained for locally convex spaces. that i f

Exi

is weak subseries convergent, then

convergent with respect to the Mackey topology

Exi

Tweddle shows is subseries

T(E,G ).

Moreover,

I

he shows that E

T(E,G )

is the strongest locally convex topology on

s u c h that every element of

3

is subseries convergent.

Although Tweddle's r e s u l t is more comprehensive than Theorem 3, Theorem 3 is general enough to have some interesting consequences. At t h i s point we present an application of Theorem 3 to the scalar version of the Nikodym Convergence Theorem for countably additive measures.

That i s ,

we show t h a t

t h e Nikodym C o n v e r g e n c e Theorem can

be viewed as an O r l i c z - P e t t i s

result.

Let

sets

: E * R

be c o u n t a b l y a d d i t i v e

of a set

i E ~.

S

and l e t

Assume t h a t

~i

lim ~ i ( E ) i

= ~(E)

S(E) be t h e v e c t o r s p a c e o f a l l all

countably additive

topology and

ca(E).

E CE

J

o(S(E), If

{Ej}

is o-subseries

exists

be a o - a l g e b r a

for each

E-simple functions

m e a s u r e s on

ca(E)) = o

E

E.

Equip

from t h e n a t u r a l

is a disjoint convergent in

S(E).

Let

ca(E)

w i t h t h e weak

duality

s e q u e n c e from

for each

E E E.

and l e t

S(E)

of sub-

between

S(E)

E, t h e s e r i e s

By Theorem 3, t h e s e r i e s

J

is subseries

c o n v e r g e n t in t h e t o p l o g y o f u n i f o r m c o n v e r g e n c e on

o(ca(Z),

$(E))

o(ca(E),

S(E))

gi(Ej)

- Cauchy s e q u e n c e s .

But,

Cauchy by h y p o t h e s i s .

c o n v e r g e u n i f o r m l y in

i,

the sequence

{gi }

T h i s means t h a t i.e.,

the

{~i }

is

the series

are uniformly

3 countably additive.

T h i s i s p a r t o f t h e c o n c l u s i o n i n t h e Nikodym

C o n v e r g e n c e Theorem (Theorem S . 9 ( i i ) ) . clusion

(Theorem 5 . 9 ( i ) )

The o t h e r p a r t o f t h e c o n -

f o l l o w s from t h i s .

be

62 Notice that in general subseries

t h e a n a l o g u e o f Theorem 2 f o r t h e weak ~ t o p o l o g y i s

false.

For example, the series

convergent but certainly

Diestel

and F a i r e s

an O r l i c z - P e t t i s logies

([33]).

generality

in s e c t i o n

less general

result

the Diestel-Faires

result

follows quickly

conditions

for

in its

full

a somewhat

from the matrix methods

point.

Theorem 4.

be a normed s p a c e s u c h t h a t

X

convergent.

t i m e we d e r i v e

e m p l o y e d up t o t h i s

Let

i s weak ~

t o t h e weak ~ and norm t o p o -

10. At t h e p r e s e n t that

~

and s u f f i c i e n t

to hold relative

We d e r i v e

in

n o t norm s u b s e r i e s

have given necessary result

Zej

X

contains

no s u b -

e

space isomorphic (topologically) to ~

convergent

Proof:

in

£I

If

Ex i

is weak ~ subseries

t

X ,

For each

then

i

Zx i

pick

i s norm s u b s e r i e s

xi E X

such that

convergent.

[ [ x i [ [ = 1 -and

e

<xi,

xi> + (1/i)

{x i}

> [ [ x i [ ].

has a subsequence

By a r e s u l t

{Xki}

of Rosenthal

([49]

w h i c h i s weak Cauchy.

2.e.5),

Consider the

w

matrix

[ zij]

= [<Xkj, X k i > ] .

series

Exj

i s weak ~ s u b s e r i e s

Since

X

that

[zij]

lim zii i

is a

= 0

X

matrix.

so that

Theorem 1 now g i v e s t h e r e s u l t .

The D i e s t e l - F a i r e s that

i s weak Cauchy and t h e

convergent,

The B a s i c M a t r i x Theorem i m p l i e s l i m [ [ x i [ [ = O.

{x i}

contains

result

mentioned above replaces

no s u b s p a c e i s o m o r p h i c t o

£1

the condition

by t h e more g e n e r a l

e

condition sider

that

this

X

contains

more g e n e r a l

We n e x t c o n s i d e r w h i c h i s due t o K a l t o n K(X,Y) that

no s u b s p a c e i s o m o r p h i c t o

result

in section

an O r l i c z - P e t t i s ([44]).

If

be t h e s p a c e o f compact o p e r a t o r s

t h e weak o p e r a t o r

t o p o l o g y on

!

norms

and

T * [ [, x fi X, y

K(X,Y) ,

We c o n -

10.

result X

£~.

f o r compact o p e r a t o r s Y

from

a r e normed s p a c e s , X

into

is generated

Y.

Recall

by t h e s e m i -

$

E Y

([381VI.

1.3).

let

Kalton's

63 result

is g i v e n in

Theorem 5.

Let

E Ti

be s u b s e r i e s

s p e c t t o t h e weak o p e r a t o r isomorphic to

~,

Proof:

First

range.

Therefore,

E Ti

since each

{Ti),

Next, observe that

Ti

Y

contains

with re-

no s u b s p a c e

convergent with respect

i s compact) i t has s e p a r a b l e

by t h e c l o s e d s u b s p a c e spanned by

we may assume t h a t the subseries

topology implies that

weak ~ s u b s ) t i e s

X

is subseries

by r e p l a c i n g

the ranges of the

If

K(X,Y)

K(X,Y).

note that

weak o p e r a t o r

topology.

then

t o t h e norm t o p o l o g y o f

c o n v e r g e n t in

c o n v e r g e n t in

X'

Y

is separable.

convergence of

in t h e

E T'j y ' = E y T j

the series for each

E Tj

y' E Y'.

Since

is

X

con-

I I

rains no copy of

~ ,

the s e r i e s

E Tjy

is actually norm subs)ties

convergent ([33] 1.2 or I 0 . i 0 ) . 0

For each

II T'j Y j [ I

j

pick

0

s u c h that

IITjll = IITjlI.

+ (l/j))

'

°

yj E Y '

[[yj[[ = 1

Since

V

and

is separable,

the

e

B a n a c h - A l a o g l u Theorem i m p l i e s t h a t

)

(yj}

has a s u b s e q u e n c e

)

which c o n v e r g e s weak ~ difficulties

later,

t o an e l e m e n t

we assume

y

{Ykj}

o

C Y .

To a v o i d n o t a t i o n a l

k i = i. ''

NOw c o n s i d e r lim z i j i

lim 1

=

T'jYi'

( [ 3 8 ] VI. 5 . 6 ) . |

exists

If

{mj}

[zij]

= [ T j y i] = [ y . T j ] .

For e a c h

in norm by t h e c o m p a c t n e s s o f i s any s u b s e q u e n c e ,

j,

Tj

the series

I

E Zim j = E TmjY i J

the matrix

c o n v e r g e s in norm f o r e a c h

i.

Moreover,

J

limiEjZimj = limiZjYi Tmj = limiYi EjTmj = limi(EjTmj) Yi exists

in norm by t h e c o m p a c t n e s s o f

[zij]

is a

lim [ [ z i i [ [ i

~

matrix.

Zj Tmj ([38] Vl. 5 . 6 ) .

Hence,

By t h e B a s i c M a t r i x Theorem,

= lim [ [ T i Y i [ [ = 0 i

so

lim [ [ T i [ [ = lim [ [ T i [ [ i i

= 0,

64 and Theorem 1 g i v e s t h e r e s u l t . There are s e v e r a l be made. tors,

First,

L(X,Y) = K(X,Y).

that

will

in t h e s p a c e

topology.

Let

T E L(X,Y),

If this

and

X

X

be a

(where f i

xi).

any s e r i e s

H TP k k

series

That i s ,

L(X,Y)

is the coefficient X

has t h e p r o p e r t y

Then f o r any

convergent for the

f o r t h e weak o p e r a t o r convergent,

T

so t h a t

In p a r t i c u l a r ,

L(X,X) = K(X,X),

topo-

the sequence of

i s norm c o n v e r g e n t t o

L(X,Y) = K(X,Y).

convergent

c o n v e r g e n t f o r t h e norm

is subseries

has the p r o p e r t y above,

unless

Pk x = < f k ' X>Xk'

i s norm s u b s e r i e s

i { ~ TP k} k=l

f o r compact o p e r a -

which is s u b s e r i e s

be t h e p r o j e c t i o n

should

B - s p a c e w i t h an u n c o n d i -

Now s u p p o s e

t o p o l o g y and, t h e r e f o r e ,

compact o p e r a t o r s T 6 K(X,Y).

hold for

topology is subseries

the series

strong operator logy.

Pk

to

L(X,Y)

i n t h e weak o p e r a t o r

let

{ x i , f i}

with respect

only valid

not g e n e r a l l y

F o r example,

Schauder basis

functional

c o n c e r n i n g Theorem S t h a t

Theorem 5 i s b a s i c a l l y

such a r e s u l t

tional

observations

and

if

X = Y

X

must be

finite dimensional. Next note that the assumption that isomorphic to

Z~

X

contains no subspace

is actually necessary in the following sense.

the conclusion of Theorem 5 holds for all normed spaces hold for the scalar field

R.

it must

But the weak operator topology of

e

K(X,R) = X

Y,

If

m

iS just the weak ~ topology of

X .

Therefore, if the

conclusion of the theorem holds, weak M subseries convergent series must be norm subseries convergent.

By the Diestel-Faires result

I

([33]),

X

cannot contain a subspace isomorphic to

complete study of this

situation

We next c o n s i d e r r e s u l t s c o n v e r g e n c e in s e v e r a l a compact

from

S

is given in [32].

of the classical

into

A more

concerning the topology of pointwise

H a u s d o r f f s p a c e and l e t

uous f u n c t i o n s

Z~ .

G,

CG(S) where

function

spaces.

Let

be t h e s p a c e o f a l l G

i s a normed g r o u p .

S

be

continThe

65

CG(S)

t o p o l o g y o f p o i n t w i s e c o n v e r g e n c e on by t h e f a m i l y o f q u a s i - n o r m s ,

f ÷ If(t)[,

is the topology induced where

t h e quasi=norm which g e n e r a t e s t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on If[ = s u p { I f ( t ) [

CG(S)

: t E S).

[[

is

The t o p o l o g y o f

C o n c e r n i n g t h e f u n c t i o n space

CG(S),

we

result.

(fi } ~CG(S).

Let

G.

and

i s g e n e r a t e d by t h e q u a s i - n o r m

have t h e f o l l o w i n g O r l i c z - P e t t i s

Theorem 6.

t E S

If

Ef i

is subseries convergent with

respect to the topology of pointwise convergence, then

Zf i

is

subseries convergent with respect to the topology of uniform convergence.

Proof: 0

From Theorem I ,

in norm.

[fi(ti)]

For e a c h

it suffices i

pick

= sup { [ f i ( t i ) [

t i E Si

{t i}

and a

t E S

(fi)

converges to

such t h a t

: t E S} = I f i l .

We make t h e f o l l o w i n g c l a i m : of

t o show t h a t

such t h a t

there exist

a subsequence

limi f . ( t j mi) = f j ( t )

(tmj}

for each

jEB. For t h i s ,

let

G!1 be t h e space o f a l l

G-valued sequences

equipped w i t h t h e q u a s i - n o r m

igl = ~ i=l where

g = ( g l , g 2 . . . . ) E G~.

F(s) = (fl(s),f2(s) is continuous.

. . . . ).

The s e t

is a metric group, t h i s a subsequence that j.

Igil

This establishes

Define

,

F : S ~ G~

Note t h a t F(S)

F

by

is continuous since each

i s compact and, moreover, s i n c e

set is sequentially

{F(tmi} o f

F(tmi) ~ F(t)

/2 i (1 + I g i l )

{F(ti)}

compact.

and an e l e m e n t

or, equivalently, t h e c l a i m above.

limx f j ( t m i )

fi

G[q

Thus, t h e r e i s F ( t ) E F(S) = fj(t)

such

for each

66 Now suppose t h a t

{[fi[}

assume t h a t t h e r e e x i s t s

e > 0

t h e c l a i m above t h e r e e x i s t such t h a t

mi = i.

ies convergence.

such t h a t

a subsequence

liml f j ( t m i } = f j ( t )

assume t h a t

doesn't converge to

Note t h a t

fj(t)

Note t h a t t h e rows and columns o f eij > 0

such t h a t

~ e i j < =" 1)j

sequence o f p o s i t i v e

i # j.

Let

f E CG(S)

In p a r t i c u l a r , there exist

integers

a subsequence

Z

liml f p j ( t q i )

= fpj(X)

for each

By L e n a {pi}

limi fPj(" t q i )

= fpj(t).

Hence,

for each {t i}

t E S

= [fj(t i) - fj(t)]. 0.

Choose

t h e r e i s an i n c r e a s i n g [ZpiPj [ < e i j

~ f p (s} = f ( s ) j=l j f o r each i. {pi } and an

fpj(X)

and

f(tqi) = fpj(t)

÷

for

for

s E S.

By t h e c l a i m above, x E S -such t h a t f(x). for each

Moreover, j,

and,

therefore,

If(x) - f(t)[ = [ ~j fpj(X) - ~3 fpj(t)[ < ~O[fpj(X) - fpj(t}] = O. By the triangle inequality, we have [fqi(tqi) - fqi(t)[ ~ j#i[fqj(tqjZ ) - fqj(t)[ + [f(tqi) - f(t)[ Z e. - + If( ) - f(x)[ + [f(x) - f(t)[ j=l lJ tqi

= j=IZe..13 + [ f ( t q i ) Thus,

[fql(" t q i )

- fqi(t)[

÷ 0

- f(x)[

and s i n c e

By

by t h e p o i n t w i s e s u b s e r -

2.1,

and

i.

To a v o i d s u b s c r i p t s

such t h a t

of j

Then we may

both c o n v e r g e t o

= f(tpi) {qi}

of

Z = [zij]

be such t h a t

j=lZf p j ( t p i )

j.

÷ 0

Consider the matrix

[fi [ ~ e {tmi}

for each

Oo

fqi(t)

÷ 0, we have

67 Ifqi(tqi)l Hence,

÷ 0

which contradicts

[fi I ÷ 0

the assumption that

and t h e t h e o r e m i s e s t a b l i s h e d .

The t h e o r e m a b o v e was e s t a b l i s h e d i s a normed s p a c e ( [ 7 7 ] easily

series

by Thomas f o r t h e c a s e when

Note that

Efi

in

CG(S)

topology of pointwise is also

S

The g r o u p c a s e o f t h e r e s u l t

is metrizable, is subseries

a b o v e was

D i a g o n a l Theorem.

Thomas a l s o shows t h a t

convergent with respect

c o n v e r g e n c e on a d e n s e s u b s e t

subseries

G

the matrix methods above

i n [70] by u s i n g t h e A n t o s i k - M i k u s i n s k i

In t h e c a s e when

it

II.4).

handle the group case.

established

[ f i [ ~ e.

convergent with respect

D

of

if a

to the S,

then

to the topology of uni-

form convergence.

The m a t r i x method e m p l o y e d in t h e p r o o f o f Theorem

6 also yields

result.

Let

D

this

be a d e n s e s u b s e t o f

c o n v e r g e n c e on norms,

If

Let

Zf i

point-wise

[fi[.

be m e t r i z a b l e

c o n v e r g e n c e on

topology,

we h a v e t h e

and l e t

D

be a d e n s e s u b s e t o f

convergent with respect D,

then

Xfi

to the topology of

is subseries

convergent

to the topology of uniform convergence.

For each

i,

pick

ti E D

By t h e m e t r i z a b i l i t y ,

converges to a [zij]

S

Concerning this

by t h e f a m i l y o f q u a s i -

o f Thomas ( [ 7 7 ] ) .

is subseries

with respect

Proof:

t E D.

result

Theorem 7.

The t o p o l o g y o f p o i n t w i s e

is the topology generated

f ÷ If(t)[,

Orlicz-Pettis

S.

D

S.

= [fkj(tki)]

point

{t i}

t E S.

is a

X

such that

[fi(ti)l

has a subsequence

It is easily matrix,

+ (l/i) {tki}

checked that

> which

the matrix

and t h e p r o o f i s c o m p l e t e d a s in

Theorem 6. I t s h o u l d a l s o be n o t e d t h a t be d e r i v e d

the general

from the metric case established

c a s e o f Theorem 6 c a n

a b o v e in Theorem 7 by

u s i n g a method o f Thomas in [7"7], p. 183 ( s e e a l s o

[38],

VI.7.6).

68 Since the metric case of the theorem is very easily

established,

method o f Thomas r e p r e s e n t s

to the proof of

an i n t e r e s t i n g

contrast

this

Theorem 6 g i v e n a b o v e . It is also worth noting that tains

the classical

case.

Let

X

Orlicz-Pettis

w i t h t h e weak ~ t o p o l o g y . t i n u o u s f u n c t i o n on

therefore,

S.

Let Then

Hx i

be s u b s e r i e s

be t h e u n i t b a l l o f

S

i s compact and e a c h Hx i

is subseries

Hx i i s s u b s e r i e s

is of interest

convergent with respect

C(S).

This implies that

u i t y was c e r t a i n l y

utilized

let

H

be a a - a l g e b r a

the

B

space of all

sequence of sets

from

holds for the space

bounded r e a l - v a l u e d

£~.

C(S),

and,

Zx i

is subser-

that

is,

the

(The c o n t i n -

in t h e p r o o f s o f Theorems 6 and 7 . )

the series

c o n v e r g e n t b u t n o t norm s u b s e r i e s

Theorem 8.

convergent

in Theorem 6 i s i m p o r t a n t .

of subsets of a set

H,

is a con-

t h e a n a l o g u e o f Theorem 6 i s

E-measurable equipped with the sup-norm.

ing r e s u l t

xi

t o t h e norm t o p o l o g y .

to note that

of the functions

equipped

to the topology

f o r t h e s p a c e o f bounded m e a s u r a b l e f u n c t i o n s ;

continuity

conver-

X

t o t h e t o p o l o g y o f p o i n t w i s e c o n v e r g e n c e in

ies convergent with respect

false

S

The s e r i e s

o f u n i f o r m c o n v e r g e n c e in

It

c a s e o f Theorem 6 c o n -

Theorem (Theorem 2) as a s p e c i a l

be a normed s p a c e and l e t

g e n t in t h e weak t o p o l o g y .

with respect

the scalar

S

and l e t

functions If

HCE. J

{Ej}

S

be

which are

is a d i s j o i n t

i s pointwise subseries

convergent.

For t h e s p a c e

on

B(S,H)

For

In p a r t i c u l a r ,

B(S,Z)

this

we have t h e f o l l o w -

for the topology of pointwise convergence.

If

[fi

is subseries convergent in

B(S,H) with respect

to the topology of pointwise convergence, then

Zf i

convergent with respect to the Mackey topology

T(B(S,H), ca(E)).

Proof:

If

a

is an i n f i n i t e subset of

~,

we write

is subseries

K fi iEa

for the

69 (pointwise)

sum o f t h e s u b s e r i e s

subsequence

{mj}.

If

o

Ef m , J J

is finite,

Define a family of countably additive on t h e power s e t

7

of

E fi E B(S,E), leo

this

[q

by

where

is ordered as the

o

t h e meaning o f scalar

E fi iEo

measures

/Lt(o) -- ( E f i ) ( t ) . i Eo

is clear.

{gt : t E S} Since each

family of measures is point-wise

bounded on

7.

By t h e Nikodym Boundedness Theorem t h e f a m i l y i s u n i f o r m l y bounded. That i s ,

the family

{i E fiEo

: 0 ~ ]~)

i s u n i f o r m l y bounded in

B(S,E). If

partial

{mj )

is any increasing sequence of positive integers, the k { E f

sums

j=l

ably additive that

mj}

a r e u n i f o r m l y bounded, so i f

measure on

E,

•

j

That i s ,

the series

Efj

is subseries

o(B(S,Z),

ca(E)).

J

convergent with respect

to the topology

version of the Orlicz-Pettis is subseries

is a count-

t h e Bounded C o n v e r g e n c e Theorem i m p l i e s

= E " j

v

Theorem f o r l o c a l l y

By t h e

convex spaces,

c o n v e r g e n t f o r t h e Mackey t o p o l o g y

Efi

v(B(S,E), ca(E))

([77]). The r e s u l t s and, sible

in a c e r t a i n

i n [71] g i v e a more d e t a i l e d sense,

for the space

study of this

show t h a t Theorem 8 i s t h e b e s t r e s u l t

0 < p < ~.

f : N ÷ G

The s p a c e

such that

(Z I f ( i ) I P ) l i p If]p = E ]f{i)J p

£P{G)

is the space of all

E i { f ( i ) [ p < ~.

defines

For

a q u a s i - n o r m on

defines

a q u a s i - n o r m on

1 ~ p < ~, ~P(G),

g e n e r a t e d by t h e q u a s i - n o r m s ,

f ÷ If(i)l

~P-spaces. sequences {f{p =

and f o r

0 < p < 1,

~P(G).

The t o p o l o g y o f p o i n t w i s e c o n v e r g e n c e on

this

pos-

B(S,E).

We n e x t c o n s i d e r t h e a n a l o g u e o f Theorem 6 f o r t h e Let

situation

for

£P(G)

is the topology

i E ~.

t o p o l o g y we have t h e a n a l o g u e o f Theorem 6 f o r

Concerning

AP(G).

70 Theorem 9.

Let

Z fi

be s u b s e r i e s

topology of pointwise gent with respect

Proof:

convergence.

O, 0 . . . .

f j E t,P(G). integers

Then

Z fi

to the topology generated

Consider the matrix

fj(i),

convergent with respect

zij

) E ~P(G).

Next l e t

and l e t

~f j=l

l im z i j l

by

lip.

= fj

in

be an i n c r e a s i n g

represent

mj

is subseries

conver-

= (fi(1)

First

{mj}

to the

~P(G)

since

sequence of positive

the subseries

sum w i t h r e s p e c t

to

the topology of pointwise convergence. Now j=lX j = (j=l"-fmj (i)'

convergence and

j=IX mj(i)' O,

) by the pointwise

lim ~- z.. = { ~lfm (k)}k=1 in ~P(G) since i j:l Ij j: j

)- fmj 6 ~P(G). Thus, [zij]

is a )[ matrix. By the Basic Matrix

Theorem, limj limi Izijlp = limj Ifjlp = O. Theorem 1 gives the result. F o r t h e c a s e when Thomas ( [ 7 7 ]

II.4

G

i s a normed s p a c e ,

and I I . 8 ) .

Theorem 9 i s due t o

The g r o u p c a s e was e s t a b l i s h e d

by e m p l o y i n g t h e A n t o s i k - M i k u s i n s k i

D i a g o n a l Theorem.

Theorem 9 m o t i v a t e s t h e f o l l o w i n g a b s t r a c t result

which is applicable

spaces as well as certain

Theorem 10. logy

a

there

exists

(i) (ii)

Let

that

X

of the classical

linear

is

lim Tix = x operator

o -

1[

for each

metric

operators

type

sequence ([74]).

space with a vector

is weaker than the original

Ti

Orlicz-Pettis

spaces having a Schauder basis

a sequence of linear

each

identity

to several

be a m e t r i c

i n [70]

topology.

Ti : X ÷ X

topoSuppose

such that

continuous x E X

(i.e.,

in the strong operator

{T i} topology).

converges to the If the series

71 Zx i

is o-subseries

convergent,

then

Zx i

is

[[-subseries

convergent.

Proof: Consider the matrix [zij] = [Tixj]. From(ii), it follows that

lim zij 1

increasing

= xj

for each

j.

sequence of positive

From ( i ) , integers

it

follows that

{mj},

~j Zim j

f o r any

converges to

Ti(~, Xmj) (where ~j Xmj is the o-limit of the subseries). Hence, (ii) implies that limij Z Zimj = Zj Xmj exists. Thus, [zi;]j is a matrix.

The B a s i c M a t r i x Theorem i m p l i e s

lim lim Tix j = lim xj = 0

j

i

(in the metric topology).

j

yields

that

the result. It

is easy to see that

Theorem 9 i s a c o r o l l a r y

the topological

vector

o f Theorem 10 by t a k i n g

logy of pointwise

c o n v e r g e n c e and by d e f i n i n g

by

) = (x I . . . .

T i ( x 1, x 2 . . . .

x i, 0 ....

).

w i s e c o n v e r g e n c e on t h e s e q u e n c e s p a c e s treated

in a s i m i l a r

Theorem 10.

Let

space version o

of

t o be t h e t o p o -

Ti :

zP(x) ~ zP(x)

The t o p o l o g y o f p o i n t -

Co(X)

and

c(X)

c a n be

fashion.

A B-space with a Schauder basis

(fi

Theorem 1 now

X

be a

is the coordinate

c a n a l s o be t r e a t e d

by u s i n g

B-space with a Schauder basis

functional

relative

to

xi).

Let

{x i , F

fi }

be t h e

e

subspace of logy

o(X,F).

X

s p a n n e d by t h e In t h e c a s e o f

the topology of pointwise Define o -

I[

Ti : X ÷ X

the strong operator

zP,

and

We c o n c l u d e t h i s

or

c,

o be t h e weak t o p o this

topology is just

convergence.

{T i}

Ti

By Theorem 10 any s e r i e s

convergent section

Then e a c h

converges to the identity

topology.

which i s o - s u b s e r i e s

co

and l e t

i T i x = E < f u , x> x k. k=1

by

II c o n t i n u o u s

{fi }

is also norm-subseries

by e s t a b l i s h i n g

is operator

~ Yi

in

convergent.

an a b s t r a c t

type of

in X

72 Orlicz-Pettis

result

which was p r e s e n t e d in [ 7 0 ] .

be normed g r o u p s and l e t s e c t i o n 6, we l e t

o(E,F)

each of the additive topology

o(F,E)

b : E x F ~ G

Let

E, F

be a b i a d d i t i v e

be t h e w e a k e s t t o p o l o g y on

maps

{b(.,y)

: y E F}

is defined similarly.)

a l s o d e p e n d s on t h e s p a c e

G

and

map. E

As in

such that

is continuous.

(The

Again t h e t o p o l o g y

and t h e map

G

o(E,F)

b, but t h e n o t a t i o n

should

c a u s e no difficulties.

Let We let

CF

be the family of all

C(E,F)

be the topology on

ments of the family net

{xv)

CF

of

E

F.

of uniform convergence on ele-

o(F,E)-Cauchy sequences in F,

E

converges to

0

in the topology

lim b(xv,f i) = 0

uniformly in

i

for each

C(E,F)

in

o(F,E)-Cauchy sequences in

is generated by the quasi-norms

~(E,F)

[fi ) 6 C F.

i.e., a iff

The topology

{X{a = sup{b(x,fi){,

a = {fi } ranges over the family of all

where

o(F,E)-Cauchy sequences.

Concerning these topologies, we have the following OrliczPettis result.

Theorem 11. o(E,F),

Proof: [zij]

If

then

Let

E xi

is subseries

E xi

{fi )

is subseries

be

= [b(xj,fi)].

convergent with respect

convergent with respect

o(F,E)-Cauchy. Since

{fi }

is

to

to

C(E,F).

Consider the matrix a(F,E)-Cauchy,

lim z i j

exists.

I

If

{mj)

i s an i n c r e a s i n g

Ej Zim j = b(Exjmj' fi ),

series. Hence,

Thus, [zij]

where

liml~J Zim j is a

lim b(xj,f i) = 0 J

~

sequence of positive

EXmj

is the

exists since

matrix.

uniformly in

integers,

then

o(E,F)-sum of the sub{fi } is

o(F,EkzCauchy.

The Basic Matrix Theorem implies that i,

i.e., xj ~ 0

in

C(E,F).

Theorem 1 now gives the result. Theorem II is established in [70] by employing the Antosik-

73 M i k u s i n s k i D i a g o n a l Theorem, and i t number o f known O r l i c z - P e t t i s as c o r o l l a r i e s . directly

results

i s shown i n [701 t h a t

a large

c a n be d e r i v e d from Theorem 11

F o r e x a m p l e , Theorems 6 and 9 c a n be o b t a i n e d

from Theorem 11.

We r e f e r

t h e r e a d e r t o [70] f o r d e t a i l s .

8.

The S c h u r and P h i l l i p s

In t h i s Phillips

Lemma

s e c t i o n we d i s c u s s

the classical

and show how t h e s e r e s u l t s

methods d e v e l o p e d i n s e c t i o n lemma a s s e r t s

that

verges strongly.

2.

f o u n d many a p p l i c a t i o n s

c a n be t r e a t e d

~1

in f u n c t i o n a l

([59]},

Phillips'

lemma h a s many a p p l i c a t i o n s

co

analysis;

14.4.0).

various abstract space versions

f o r example, there

it conf o r m s have

including the original

proof

Similarly,

i n b o t h m e a s u r e t h e o r y and

the original

application

i s no c o n t i n u o u s p r o j e c t i o n

Both o f t h e s e r e s u l t s settings.

Schur

f o r example, many o f

u s e t h e S c h u r lemma i n some f o r m .

showed t h a t

([60]

more g e n e r a l

analysis;

Theorem,

of Pettis

Phillips

converges weakly i f f

and some o f i t s

the proofs of the Orlicz-Pettis

functional

by t h e m a t r i x

One v e r s i o n o f t h e c l a s s i c a l

a sequence in This result

lemmas o f S c h u r and

of

of ~"

onto

have been g e n e r a l i z e d

to

F o r example, B r o o k s h a s g i v e n Banach

f o r b o t h t h e S c h u r and P h i l l i p s "

lemmas ( [ 2 3 ] )

and

R o b e r t s o n has g i v e n a g r o u p v e r s i o n o f t h e Schur lemma ( [ 6 2 ] } . this

s e c t i o n we p r e s e n t

Phillips

lemmas.

group-valued versions

As an a p p l i c a t i o n

give a generalization

In

o f b o t h t h e S c h u r and

o f o u r g e n e r a l S c h u r Iemma, we

of another classical

result

in summability

t h e o r y w h i c h i s a l s o due t o S c h u r . We f i r s t

establish

then indicate alization

how t h i s

i s an i n f i n i t e

elements of

Z xj jEo o

o f t h e S c h u r lemma and

can be l e g i t i m a t e l y

N

G

will

and i f

d e n o t e a normed g r o u p .

Zx.

jJ

is subseries

f o r t h e sum o f t h e s e r i e s

a r e a r r a n g e d in t h e s u b s e q u e n c e

t h e meaning o f

viewed as a g e n e r -

S c h u r lemma.

section,

subset of

we w r i t e

finite,

result

of the classical

Throughout this

G,

our g e n e r a l i z a t i o n

E xj

jEo

is clear.

~ x , j = l nj {nj}.

If

a

c o n v e r g e n t in

where t h e If

a c ~

is

75 Theorem 1. matrix

Let

[xij]

xij E G

for

are subseries

i,j

E ~.

Assume t h a t t h e rows o f t h e

c o n v e r g e n t and

lim x i j

= xj

exists

for

1

each

j.

If

(i)

i s c o n v e r g e n t in

jEo

the series

(ii)

Proof:

{ ~ xij} Exj

o c ~.

First

the sequence

( Z xij}

we show t h a t

uniformly with respect 6 > 0

to

sup I [ (x n

0

jEo

Zij = Xni+l j - Xni j

oI

such that

of

{zij)

lJEolZml j Z

go to

N1 Z I zij I < 6/2 j=l such that T1

0, for

and

i ) m 2.

I Z z m ;I > 6. jEo 2 2 J and

T2

{n i}

such t h a t

Set

then

a Cauchy

i s not t h e c a s e ,

;-x n ;) I > 6.

i+l J

Set

there is an

satisfies

If this

m I = I.

I > 6.

jEo

o ~ ~.

and a s u b s e q u e n c e

(I)

o E ~,

c o n v e r g e n t and

uniformly for

there is a

Note

for each

lim E x i. = Z xj i jEo J jEo

condition

Set

is subseries

G

iJ

By (I) there exists a finite N I = max o I.

m2 > m I

Since the columns

such that

Again by (1), there is a finite T1 = oI

and

T 2 = 0 2 \ {j:I~j~NI}.

are disjoint with N1

max T 1 < m i n r 2

and

IOET2ZZm_jlz ) [JEo2ZZm2 o.l -j=l [ IZm2Jl • 6/2.

Continuing this construction produces a subsequence disjoint finite sets (2)

{T i}

(m i}

and

satisfying lj~riZmiJl • 6/2.

NOW c o n s i d e r t h e m a t r i x

[Yij]

02

= ~[uc~rjZmik]" The columns of

76 {Yij}

go to

O,

and if

subseries of the series matrix.

Hence,

{pj}

is any subsequence,

E Zmi ~ J

lim Yii = O. i

Thus, the matrix

~ yin vj

ji=l

is a

{yil)j is a

X

But this contradicts (2).

We now establish ( i ) and also

• x_

(3)

j=l nj

for any subsequence established,

{n j).

Let

there exists an

(4)

N

= l i m ~ x. , i j=l ,nj E > O.

By what we have just

such that

lj~o(xij-Xkj)[ < El3

for

i , k ~ N.

M

Hence, for each M and k > N, [j=l )- (Xnj-Xknj)l ( E/3.

M

{5)

~

I E X.

j=l " j

Thus

M

-E

x- [ ( I E ix n -Xknj) I + I E ix,.. -XN. )l j=l Icnj j=1 j j=M÷I " " j "j

+ [ ~ x- [ < 2E13 + I ~ XNnj[ j=M+INnj j=M÷I and the last term on the right hand side of iS) is small for large.

M

This establishes (3).

From (3) and the uniform Cauchy condition {4}, it follows that lim Z xi: = E xj i jEo J jEa

uniformly for

a c ~

and the proof is complete.

Theorem I generalizes the version of the Schur lemma for spaces as given by Brooks in [231, Corollary 2.

B-

Brooks methods do

not generalize to non-locally convex spaces or groups since they depend upon duality methods.

Robertson has established a (more gen-

eral) form of Theorem 1 in [62]; his methods are quite different than the matrix methods employed above and depend heavily on Baire category methods.

77 We now d e r i v e dicating

that

exists

S c h u r lemma f r o m Theorem 1 t h u s

Theorem 1 c a n be v i e w e d a s a l e g i t i m a t e

of the classical

Corollary

the classical

2.

generalization

S c h u r lemma.

(Schur) Let

for each

in-

o E ~

be a r e a l

[tij]

matrix.

t j = lim t i j ,

and i f

If

then

lim Z ti: i jEo J

( t j ) E C1

I

and

lim Z

i j=l Itij

Proof:

Let

- tjl

e • O.

I Z (tij-tj)l jEo

By Theorem 1, f o r l a r g e

< e for

The u s u a l

statement

1.3.1).

let

mo

finite

more g e n e r a l

in [ 2 5 ] .

be t h e s u b s p a c e o f

sequence

The h y p o t h e s i s

{x i }

conclusion

C~

which c o n s i s t s

in C o r o l l a r y

= 0

h a s b e e n g i v e n by

E C1

for each

C1

this

that

i s norm c o n v e r g e n t ;

lemma i s s o m e t i m e s s t a t e d F o r u s e in s e c t i o n 1 c a n be s t r e n g t h e n e d that

([79]

the series

The

any w e a k l y c o n v e r g e n t the Schur

14.4.7). the conclusion

somewhat when t h e m a t r i x h a s v a l u e s a series

and

norm c o n v e r -

i s one way t h a t

g we a l s o n o t e t h a t

s a i d t o be bounded m u l t i p ! i e r o f bounded s c a l a r s ,

this

i

the

o(~l,mo).

is actually

implies that

of

of the sequences with

2 is just

{x i )

in Corollary

space interpretation

x i = {tij}

the sequence

In p a r t i c u l a r ,

Recall

form

The f u n c t i o n Let

tj

i s a Cauchy s e q u e n c e in t h e t o p o l o g y

is then that

C1.

sequence in

space.

o f t h e Schur lemma h a s

2 is the following.

range.

gent in

But then for such i ,

This s)ightly

B r o o k s and M i k u s i n s k i Corollary

o c ~.

i,

({61] 1 " 1.2) "

j=l [tij-tj [ ( 2 e

2 ([79]

= O.

Zx i

in a topological

convergent Ztix i

vector

in Theorem i n a Banach space is

if for each sequence

is convergent.

{tj)

In a B - s p a c e

78 subseries lent

c o n v e r g e n c e and bounded m u l t i p l i e r

([63]

III.6.5).

conditions

For series

convergence are equiva-

in B - s p a c e s w h i c h s a t i s f y

in Theorem i , we h a v e t h e f o l l o w i n g

the

u n i f o r m bounded

multiplier result.

Corollary,3. hypothesis

Let

X

be a B - s p a c e and

o f Theorem 1.

Let

e > O.

satisfy

the

Then

limij=l ~ tj.xij --j=l ~ tj.x.j uniformly for

Proof:

xij E X

ll(tj)ll (

{tj) E ~,~ with

By Theorem 1, t h e r e

exists

N

such that

I.

i ) N

g

implies

[I Z ( x i j - x j ) [ l

jEo

i

< e

for

o c ~.

Thus, f o r

x

fi X

and

e

I[x II ( i,

IjEo Z < x

'xij-xj >

I < e

for

i ) N

j=l~ [ < x ,xij - xj > [ ( 2e

for

i ) N,

and

o c ~.

This

implies that )

(6)

([611 1.1.2).

Now for

II j=l ~t.(xi,-x.)ll J a J

{tj} E £~

= sup{I <x'

sup{j=1 ~ Itjl for

)

llx II ~ 1

[tj[ ( 1,

with

~tj(xij-xj)

we have from (6)

> I : IIx'll

'j=l

I < x', xij-x j > I : Ilx'll

(1)

(1}

( 2e

i ) N. Corollary 3 has the following interesting operator interpreta-

tion.

Each subseries convergent series

linear

operator

T i : £~ ÷ X

o f Theorem 1 i m p l i e s

that

of Corollary

induces a bounded

Ti{t j) = Z tjxij. 3

the sequence

for the topology of pointwise The c o n c l u s i o n

by

E×.. j ij

{Ti}

The h y p o t h e s i s

i s a Cauchy s e q u e n c e

c o n v e r g e n c e on t h e s u b s p a c e

3 is then that

there

exists

mo o f

a bounded

£®.

79 linear

operator

series

Zxj

T

{T i}

the subseries

actually

o r norm t o p o l o g y o f

Corollary

classical

by

induced

and t h e s e q u e n c e

uniform operator tation,

is

which

convergent

converges to

L(~,X).

With t h i s

Recall

that

(~,c)

((mo,C))

a scalar

matrix

of this

Ax

c

i.e.,

if for each

is well-defined

A

sufficient

t o be o f c l a s s

and p r o d u c e s a The c l a s s i c a l

Let

(~=,c)

condition

xij

such that

(mo,C) ( C o t .

Then t h e s e r i e s

I Z xi

I < e

for all

i

when

if

Ex i •

j

a

e > 0, t h e r e

min a ~ N.

j

By the uniform Cauchy condition of Theorem I, there is an M

such that xij

or

t o Theorem I .

be as in Theorem 1.

jco

Proof:

summa-

t o m a t r i c e s w i t h v a l u e s in a g r o u p .

a r e u n o r d e r e d u n i f o r m l y c o n v e r g e n t in t h e s e n s e t h a t N

x E ~= (x E mo),

We u s e Theorem 1 t o g i v e a g e n e r a l i z a t i o n

Namely, we have t h e f o l l o w i n g c o r o l l a r y

exists

the se-

o f Hahn and S c h u r g i v e s n e c e s s a r y and s u f f i f f i e n t

for a matrix

4.

( { t j } E mo),

(§ 5 o r [50] § 7 ) .

[68] o r [50] 7 . 6 ) .

Corollary

5.13).

i s s a i d t o be o f c l a s s

x = {tj} E ~

is convergent,

sequence belonging to

conditions

A = [aij]

if for each

{~ a i j t j } i J

result

of a classical

w h i c h i s a l s o due t o S c h u r ( s e e C o r o l l a r y

the formal matrix product

5.15,

interpre-

S c h u r lemma ( C o r o l l a r y 2 ) .

summability result

bility

in the

3 c a n be viewed a s a v e c t o r v e r s i o n o f t h e

We n e x t u s e Theorem 1 t o d e r i v e a g e n e r a l i z a t i o n

quence

T

] j6o H ( xij-Xkj) [ < e / 2

is subseries

convergent,

for there

i , k > M and o c ~. i s an

N

Since each

such t h a t

J Ij~axij I < e/2

whenever

1 ~ i ~ M and

min o ~ N.

Hence, f o r

min o ) N and

i ) M, lj6oZxij] ( lj6oZ(xi.-XM.)Ij j + lj~oXMjl < e

80 and the result follows.

For scalar matrices

A = [aij],

the conclusion of Corollary 4

is equivalent to the sufficient condition (i) in Theorem 5.14. if the conclusion of Corollary 4 holds for ([61] I.I.2)

A,

then

For,

~ ]ai ] ~ 2e j)N J

and this is just condition 5.14 (i) in the classical

Schur summability result.

Thus, Theorem 1 and Corollary 4 yield

another proof of the ~ h u r

summability result given in 5.15.

We can also use the r e s u l t s in Theorem 1 and Corollaries 3 and 4 to consider a vector version of the c l a s s i c a l summability r e s u l t s above.

Let

X be a B-space and let

valued sequences

(xj}

which are convergent.

i n f i n i t e matrix with values in class

(mo,c(X)) ( ( ~ , c ( X ) ) )

the sequence

{ ~ tjxij} i 3

matrix product

Ay

c(X) be the space of a l l

X,

then

c(X).

i s an

y = {tj} E mO ({tj} E ~ ) ,

c(X),

is well-defined for each

produces a sequence in

A = [xij ]

A is said to be in the

i f for each

belongs to

If

X-

i . e . , if theformal y E mo (y E ~ )

and

Using the r e s u l t s above, we have the

following vector summability r e s u l t which gives a vector version of the c l a s s i c a l Hahn-Schur summability r e s u l t s (5.15).

Theorem S.

Let

A = [xij]

subseries convergent. (a)

A E (~,

(b)

A E (mo, c(X))

(c)

(i) the series

be an i n f i n i t e matrix whose rows are

The following are equivalent:

c(X))

~x. j lj

are unordered uniformly convergent

(see Cot. 4) and (ii) lim xij = xj

exists for each

j.

i

(d)

lim Z x.. = Z xj i jEa ij jEo

uniformly for

a c ~.

81

Proof:

That ( a )

Corollary (b)

3.

implies

implies

Corollary (c)(ii)

Let

N

implies

and l e t

o(n)

and (b) i m p l i e s (b) i m p l i e s

(a) by

(c)(i)

and c l e a r l y

By ( i )

there

e j 6 mo.

(d).

llj~xijll

such that

o ~ ~

4 shows t h a t

by c o n s i d e r i n g

We n e x t show ( c ) exists

(b) i s c l e a r

Let

< e

= o fl ( j

e > O. i

for all

and

min o ) N.

If

: j ~ n}.

q

o = (n i : n i < ni+ 1, i E ~} by ( i i ) . verges. sum

Hence, s i n c e That i s ,

Z xi leo

i ~ M.

B-space,

a

1=p

the subseries

is subseries

Zx i

U.Z Xn.U ( e

q > p ) N, t h e n

i

ZXmi con-

convergent,

and t h e

Then

N

( z I1× i j=l

j-xj]

I + II

z xi,ll+ll jEo(N) J

z x.ll jEo(N) J

< 3e

Hence (d) h o l d s .

That (d) i m p l i e s In a v e r y s i m i l a r logues of the classical m a t r i c e s whose e n t r i e s linear

is

the series

exists.

II z ( x i j - x j ) l l jEo for

X

and i f

spaces.

(b) i s c l e a r . fashion,

one can f o r m u l a t e and c o n s i d e r a n a -

summability results a r e bounded l i n e a r

o f Hahn and S c h u r f o r operators

Such a n a l o g u e s o f t h e c l a s s i c a l

between m e t r i c

summability results

a r e o b t a i n e d i n [76] and a r e b a s e d on t h e m a t r i x methods o f t h e s e notes. We c o n c l u d e t h i s lemma.

First,

the Phillips

Theorem 6.

section with a discussion

of the Phillips

we g i v e a s t a t e m e n t o f t h e c l a s s i c a l lemma ( [ 6 0 ] ,

(Phillips

scalar

version of

[79] 1 4 . 4 . 4 ) .

[601)

Let

Y

be t h e power s e t o f

~ and

let

82 /~i : ~ ~" •

be bounded and f i n i t e l y additive.

e x i s t s for each

E c_ ~, then

If

lim p.i(E) = ~t(E) i

lim ~ I g i ( j ) - ~(J)l = O. i j=1

Theorem 6 is often stated in t h i s form with

~(E) = 0

([60],

[ 791 ).

Theorem 6 h a s t h e f o l l o w i n g dual of

£~

functions

is the space,

on

the series

~

ba,

of all

with the total

Zu(j)

function

space interpretation.

bounded,

variation

is absolutely

finitely

norm.

convergent,

additive

For each

and

The set

u E ba,

u ÷ {u(j)}

J defines that

a projection

P

if the sequence

o(ba,

mo),

particular, respect

from

(~i}

ba

in

onto

ba

the projection

is sequentially

o(ba,

Recall

mo)

that

~ : :r ~, G

P

sequence

finitely

additive Z/~(E i )

if

{E i} set

i s norm c o n v e r g e n t

in

£1.

In

continuous with

and norm t o p o l o g i e s .

)-

is strongly

disjoint

series

i s Cauchy in t h e t o p o l o g y

{P~i}

We now g i v e a g e n e r a l i z a t i o n

then

Theorem 6 t h e n a s s e r t s

then the sequence

to the

measures.

t. 1.

o f Theorem 6 t o g r o u p - v a l u e d

is a o-algebra additive

from

function

)-

iff

(§5).

/~

of subsets

of a set

l i m ~(E i ) = 0 If

G

is strongly

converges for each disjoint

for each

is complete, additive

sequence

S,

iff

a the

(E i} ( [ 3 4 ]

1.1.18).

Theorem 7. strongly

Let

G

additive.

be s e q u e n t i a l l y If

then for each disjoint lim Z ~i(Ej) i jEo

= Z g(Ej)

jEo

lim ~i(E) sequence

c o m p l e t e and l e t = ~(E)

{Ej}

uniformly for

exists

~i

:-Z ) G

for each

be

E E Z,

from o c ~.

(In particular,

is strongly additive.)

Proof:

By Theorem i ,

for each

o E ~.

i t s u f f i c e s to show that

lim Z gi(Ei) i jEo

F i r s t consider the case when ~ = 0.

exists

We cl~aim that

83 in this

case

l i m )- ~ i ( E ; ) i jeo J

= 0

for

each

o.

If this

is

not the

case,

{0

there

is

converge

a disjoint to

0.

sequence

Thus,

there

(Ej)

such that

exist

5 > 0

( ]r ~ t i ( E ~ ) } i j=l J

doesn't

and a subsequence

{k i }

¢¢J

such that

] E gk (Ej)] j=l i

• 5.

For convenience

assume

k i = i.

Now

n1 there

exists

n 1,

n1 )- [ g i ( E ; ) [ j j=l

that

such that

< 5/2

for

I E gl(E~)l j=l J

i ) m1.

n2

• 5.

There exists

n2

[.)- ~m (Ej)[ > 5. 3=1

n2 > n 1

m1

such

such that

n2

[ ~" +~iml(Ej)[ ) [ Z gm (E~)[ -

Hence,

1

There exists

j=n 1

j=l

1

u

n1

)- lgml(Ej) [ > 5 / 2 .

j=l

Continuing

this

construction

produces

such that

ni+l ] 7" gm (E~)[ • 6/2. J=ni+l i J

is a disjoint sequence in Consider the matrix [zij]

is a J[ matrix.

zii ÷ O. the

result If

sequence

that

But

{Ej},

Put

and

there

{m i }

ni+l Fi = U Ej. J=ni+l

and

{n i }

Then (Fj}

[~mi(Fi)[ • 6/2.

[zij] = [~tmi(Fj)].

By Drewnowski's Lemma,

Hence, by the Basic Matrix Theorem,

]zii I = [gmi(Fi) I ) 6/2.

in the case

g # 0

E with

subsequences

when

lim

g = 0.

~ ~i(E;)

i j=l

exist

This contradiction establishes

J

fails

5 > 0

(Ej) - gk (Ej)){ > 5. { Z (gk j=1 i+l i

to exist

for

and a subsequence

some d i s j o i n t

{k i }

such

Applyingthe f i r s t part to the

84 sequence

ui

=

~ki+l - ~ki

gives the desired contradiction.

We now show that the c l a s s i c a l P h i l l i p s lemma (Theorem 6) f o l lows as a c o r o l l a r y of Theorem 7.

Proof of Theorem 6:

Let

{ E ( ~ i ( j ) - ~(j)){ < e jEo

j=l

]~i(J) - ~(J)l ~ 2e

• > O.

for

i

for large

By T h e o r e m 7,

large.

i

But then

([56] 1.1.2).

Theorem 7 can also be derived from Theorem I Brooks-Jewett r e s u l t . Theorem S.6 (see [14]). is taken from [14].

by employing the

Most of t h i s section

9.

The S c h u r Lemma F o r Bounded M u l t i p l i e r

Convergent Series

In this section we consider bounded multiplier convergent series in a metric linear space and present a version of the Schur lemma for such series.

Our result is based on the strengthened version of the

Schur lemma for Banach spaces which was given in Corollary 8.3. Throughout this section A series

~xi

the series scalars

in

~tix i

(ti}.

X will denote a metric linear space.

X is said to be bounded multiplier convergent i f is convergent in

A series

Hxi

X for each bounded sequence of

which is bounded multiplier convergent

is c l e a r l y subseries convergent (take

ti = 0

or

eral, the converse of this statement is false.

i s e a s y t o g i v e an example o f a s e r i e s but not bounded m u l t i p l i e r Example 3 . S . k.

~ej

That i s ,

[[

pick

[[-subseries

convergent since,

an ( n o n - l o c a l l y

{~k } E ~1 mo

by

In a normed space i t

which i s s u b s e r i e s

such that

[[(ti}[[

~k # 0

= E [ti~i[.

mo.

the series

E(1/j)ej

for each The s e r i e s

doesn't

con-

R o l e w i c z g i v e s an example o f a s e r i e s

c o n v e x ) F - s p a c e which i s s u b s e r i e s

bounded m u l t i p l i e r

convergent

c o n v e r g e n t but i s not bounded m u l t i p l i e r

in particular,

v e r g e t o an e l e m e n t o f

but, in gen-

c o n v e r g e n t by u s i n g t h e normed s p a c e i n

Then d e f i n e a norm on is

I),

convergent ([63]

III.6.9).

in

c o n v e r g e n t b u t not

In a l o c a l l y

convex

F-space a series is subseries convergent i f f i t is bounded multiplier convergent ([63] I I I . 6 . 5 ) . We f i r s t establish two preliminary lemmas. The f i r s t is an elementary property of the scalar multiplication in a metric linear space which is an immediate corollary of 6.6.

Lemma 1.

Itl ~ 1 .

If

lim x j = 0 J

in

X,

then

lim I t x j l J

= 0

uniformly for

86 See a l s o

[80]

1.2.2

for a proof.

The n e x t lemma i s a s p e c i a l which will

Lemma 2.

be e s t a b l i s h e d

Let

xij

6 X

of the general

case

S c h u r lemma

i n Theorem 3.

for

i,j

fi [q

be s u c h t h a t

]Zxi •

j

i s bounded

J

cm

multiplier

convergent

for each

i.

If

l i m ~" t . x i .

i j=l J

= 0

J

for each

mo

{tj} fi 9.~,

Proof:

then

lim )- t . x i . = 0 i j=l J J

uniformly for

If the conclusion f a i l s to hold, we may assume (by passing to

a subsequence i f necessary) that there is a sup{]j~ltjXij)

exists

: ]tjl

~ I) > 6

a I = {tlj} E ~

There exists each

ll{tj}[l ~ I.

for each

such that

6 • 0

i.

Set

such that i I = 1.

llall I ~ 1 and

M1 MI such that I X tl.X. I • 6. j=l J llJ

]j=l X tl'Xij 1j-I > 6.

Since

lim xij = 0 i --

j , by the observation in Lemma 1 above there exists

such that Now t h e r e

i ) i2 exists

M2 I X t2;x; ;I • 6. j=l J "2 J

implies a 2 = {t2j)

MI Z I < 6/2 j=l]tjxij £ ~

There exists

such that

M 2 > M I,

for

Then t h e r e

i2 • i 1

[tj I ~ 1 .

]]a21 ] ~ I

such that

and

M2 lj=1 t2'x j i 2j-I

Note IJ=MI+IZt2jxi2Jl ) [j=IZt2.x i j 23"[ -j=IZIt2jxi2jl • 6/2. Continuing this construction inductively gives subsequences {i k}

and

{Mk)

of positive integers such that

for

•

6

87

Mk { E tkjXikJ{ J=Mk_l+l

(I)

where

• 6/2

for all

k,

M_ = O. 0 Now c o n s i d e r

limj x j i = 0

the matrix

for each

i,

We c l a i m t h a t

[Zkp]

is a

{ s j } 6 £~

sj = tpj

otherwise. hypothesis.

by

[ Zkp] =[

limk ZkP = 0

if

K

for each

matrix.

Let

p

o ~ ~.

Mp_ 1 + 1 ~ i ~ Mp and

Then p~oZkp = ~ s . x i . j=l J k J Thus, the matrix

Matrix Theorem implies that

converges to

[Zkp]

is a

lim Zkk = 0. k

We now p r e s e n t our S c h u r - t y p e convergent series.

Mp Z 1. j=Mp_l+ltpiXipj

0

Since

since

I t p i { ~ 1.

Define a sequence j 6 o as

and

k * ~

sj = 0 by

K matrix and the Basic But t h i s contradicts ( I ) .

lemma f o r bounded m u l t i p l i e r

Note that conclusion (ii) in the theorem below is

just the conclusion of the Schur lemma for B-spaces given in 8.3.

Theorem 3.

Let

x i j £ X for

i,j 6 ~

be such that

bounded multiplier convergent for each exists for each (i) (ii)

Proof:

(tj} E ~ .

the series

Exj

If

i.

Exi .

ja

Assume that

lim xij : xj, i

is

lim E t

i j=l jxij

then

is bounded multiplier convergent and

lim ~ t . x i . = E t . x . ij=l J J j=iJJ

uniformly for

First we claim that the sequence

Cauchy condition uniformly for case, there exist a subsequence

{ )- t x i } j=l a a

{{{tj}]] ( I. {i k}

{{{tj}{{ ( I.

and a

satisfies a

If this is not the 5 • 0

such that

88 CO

sup{I

(2)

z

t,

j = l J (Xik+lJ

Consider the series

-

XikJ}

I : I I{tjlll

H(x i .-x i .) j k+l J kJ •

< 1) > 6.

This sequence of series

satis-

CO

fies

t h e h y p o t h e s i s o f Lemma 2 so t h a t

uniformly for

I]{tj}ll

< 1.

lim )- t . ( x . ) = 0 k j=l J Zk+lj-Xik j

This contradicts

(2) and e s t a b l i s h e s

the claim.

We now e s t a b l i s h (3)

Let

N

(i)

and a l s o

limi j =~ l t ' Jx i " J =j=l ~ tjxj

for each

By what has been e s t a b l i s h e d

£ > O.

such t h a t

i,k ) N

implies

{ t j } E ~co. above, t h e r e e x i s t s

I j6o E t .j ( x i j - X k j )1 < 6

an

for

M

a ~ ~.

Hence, f o r e v e r y

Thus, f o r

M and

i ) N I Z t,(xij-xj)l j=l J

~ 6.

i ) N, M

(4)

~

M

[ Z t~x; - ~ t.xi. ] ~ I Z t~(xj-xij)] j=l J J j=l J J j=l J CO

+ I j=M+I ~ t j ( x N j - X i J )[ + I j=M+I Z ~ 1 t j x N j I < 26 ÷ I j=M+ltjXNj The l a s t for

N

t e r m on t h e r i g h t fixed.

hand s i d e o f (4) goes t o

Condition (i)

Condition (ii)

0

and (3) f o l l o w from t h i s

as

M ÷ co

estimate.

f o l l o w s by a p p l y i n g Lemma 2 t o t h e s e r i e s

~(xij-xj). J Theorem 3 has as a c o r o l l a r y

a generalization

of another result

o f S c h u r on s u m m a b i i i t y which was d i s c u s s e d p r e v i o u s l y and 8 . 5 .

Let

A = [aij]

be a r e a l m a t r i x such t h a t

in

5.15,

the sequence

8.4

89 ( ~ t.a i.) j=l

j

The ~ h u r

A

is of class

'

summability result then asserts that the series

converge uniformly in

This condition )-t.a.. j J IJ

i.e. '

(Ao~ c). Z {aij { J

( t j ) E g=

is convergent for each

J

(for real

i

series)

( C o r o l l a r y 5 . 1 5 o r [SO] 7 . 6 ) . clearly

are uniformly convergent for

implies that the series

II{tj}ll

( 1

and

i E i~.

Using Theorem 3 we obtain the analogous result for bounded multiplier

convergent series

Corollary

4.

the series

Proof:

Let

convergent, II{tj}tl

xij

Zt.x i. jJ J

First

~tjxj J

II{tj}ll

Yx. jJ

( 1

Then

and

i E ~.

i s bounded m u l t i p l i e r

converge uniformly for

) By Theorem 3, t h e r e e x i s t s

I ~- t ~ ( x i ~ - x ~ ) l j=l J J J

above, there exists

Hence,

if the series

o f Theorem 3.

( T h i s f o l l o w s from Theorem 3 but i s a l s o e a s i l y

6 > O.

{ ~- t . x i . { j=m j J

the hy~thesis

converge uniformly for

then the series

checked directly.

implies

X.

satisfy

note that

( 1.

Let

in

< E,

< E

for

M such t h a t

{ )" t . x . { j=m j j

< E

for

))(tj}))

N

such t h a t

( 1.

i ) N

By t h e o b s e r v a t i o n

m ) M implies

1 < i (N,

II(tj)l)

( 1.

m ) M implies

~

I ~ t.xi.l ( l t~(xij-xj)l + I ~ t~x;l < 2~E j=m j J j=m j j=m j J for

i ) N,

ll(tj)ll (

Corollary

1,

and t h e r e s u l t

follows.

4 can be used t o o b t a i n a c h a r a c t e r i z a t i o n

matrices of class

(~=, c ( X ) )

when

X

is a metric linear

of space (see

90 Recall the matrix

8.5).

if the sequence

A = Ixij]

{ ~ tjxij} J

(tj} £ £co ({tj} fi mo).

is of class

is convergent in X

(£co,c(X))((mo,c(X))) for each sequence

From Corollary 4 we have the following vector

version of the Schur summability theorem for metric linear spaces.

Corollary

5.

Let the matrix

bounded m u l t i p l i e r

(a)

A = [xij]

covergent.

be such that the rows are

The following are e q u i v a l e n t :

A £ (£®,c(X))

(b)

(i)

Yt .x i • converge uniformly for

the s e r i e s and

(ii)

jJ

I I{tj}l{

limi x i j

J

< 1

= xj

exists

for each

co

(c)

lim ~ t . x . . i j=l J 1J

Proof:

exists

i m p l i e s (b) ( i i )

Let

~ I.

f o l l o w s from C o r o l l a r y

f o l l o w s by s e t t i n g

Suppose (b) h o l d s .

ll(tjlll

uniformly for

That ( a ) i m p l i e s (b) ( i )

j.

4.

That ( a )

{tj} = ej.

e > 0.

By ( i )

there exists

N

such

co

that

I Z t.xi.l j=N J J

that

{ g t.x.l j=N J J

M > 0 such that

Thus f o r

j

and ( c )

< e

1j

j

holds.

for

for

i 6 ~

ltjl

N-1 j =z l l t j ( x i j - x j

i ) M and

I ~ t.(x..-x.)l j=l

< e

ltjl~

and

( I.

{tj{ ~ i .

By ( i i )

)l < ~ for

Then ( i i )

implies

and Lemma 1 t h e r e e x i s t s

Itjl

~ 1 and

i ) M.

1,

N-1

j=l }- { t j ( x i j - x j ) {

¢o

co

+ I j=NZt~x~-,~1 + I j=NZt .Ixj . j

< 3e,

91 That ( c )

implies

This result

(a)

a l o n g w i t h t h e c o n c l u s i o n o f Theorem 8 . 5 s u g g e s t s

the following question. A

also of class

(Theorem 8 . 5 ) . spaces.

is clear.

If the matrix

(~,c(X))?

If

We show t h a t

Of c o u r s e ,

X

this

(~®,c(X)),

Let

but not X

is a B-space,

(mo,c(X)),

this

convergent,

is

is the case linear

i f t h e rows o f t h e m a t r i x a r e o n l y s u b s e r i e s

con-

t h e m a t r i x c o u l d not

but we g i v e an example o f a m a t r i x whose

rows a r e bounded m u l t i p l i e r

(mo,c(X))

is of class

i s not t h e c a s e f o r m e t r i c

v e r g e n t and not bounded m u l t i p l i e r be o f c l a s s

A

c o n v e r g e n t and w h i c h b e l o n g s t o

(~,c(X)).

be a metric linear space containing a series

Hxj which

is subseries convergent but not bounded multiplier convergent.

(For

an example in a normed space, see the example given in the introduction to this section; for an example in a complete metric linear space see [63] III.6.9.) xij = xj

if

1 ~ j ~ i

Set and

xij = 0

if

j • i.

is subseries convergent, and for each

i,

the sequence

{xij}j= 1

so that

is

lies in a finite dimensional subspace of bounded multiplier convergent. Ex., j J

we have

A = [xij]

lim E x i. = Z xj i jEo J j~o

is of class

X

Then

Hx.. j ij

Hx.. j *J

By the subseries convergence of for each

(mo,c(X)).

o c N; -

But, since

that is,

Ex. jJ

is not

bounded multiplier convergent, condition (i) of Theorem 3 fails to hold; that is,

A

is not of class

(~,c(X)).

Actually, Theorem 8.5 and its proof shows that (mo,c(X))

A

belongs to

iff condition (c) or (d) of Theorem 8.S holds.

This

observation along with Corollary S give characterizations of the classes

(~,c(X))

and

(mo,c(X))

in the case when

X

is a metric

linear space. The example given above also shows that the hypothesis in

92 Theorem 3 c a n n o t be r e p l a c e d exists

for each

type of series

by c e r t a i n

Rolewicz in [63]. space

X

scalar

sequence

generated

sequences of scalars

Rolewicz calls

a C-series

{t i} E c o .

Zxj

Ztix i

converges in

(This is a slight III.8).)

These series

Banach s p a c e

X

convergent

iff

X

contains

co

(The s e r i e s

in

co

question

that

series. valid

We f i r s t

for C-convergent

Proposition

6.

i.

for each

If

note that

Let

Proof:

Let

be s u c h t h a t

lim Z tjxij i j=l

but not o f Theorem

for C-convergent

then the series

Zxj

Zx.jl j

exists

is indeed

is C-convergent

for each

(tj}

E co .

is C-convergent.

N o t e that for each

{sj} E ~=,

the sequence

Therefore, Theorem 3 can be applied to the s e r i e s

Whereas t h e a n a l o g u e o f Theorem 3 ( i ) it

isomorphic to

in the light

Et.x. • and condition (i) implies that the series j j Ij'

series,

a

(subseries)

t h e a n a l o g u e o f Theorem 3 ( i )

xij E X

{tj} E c o .

{ s j t j ) E co .

for each

series.

Assume t h a t

xj = lim xij, i

linear

from the

is

is C-convergent

3 i s w h e t h e r t h e a n a l o g u e o f Theorem 3 i s v a l i d

by

i s known t h a t

every C-series

arises

a

have been stud-

no s u b s p a c e ( t o p o l o g i c a l l y )

Ze i

A natural

that

X

departure

in t h e c a s e o f normed s p a c e s and i t has the property

by m u l t i p l y i n g

in a metric

ied in detail

convergent.)

l i m Z x. i jEo l j

is also discussed

a series

if the series

terminology of Rolewicz ([63]

([21])

that

o c ~.

Another interesting given series

by t h e w e a k e r h y p o t h e s i s

is easy to see that

not h o l d f o r s u c h s e r i e s .

Ztjxj is convergent.

does hold for C-convergent

t h e a n a l o g u e o f Theorem 3 ( i i )

F o r an e x a m p l e ,

let

ek

does

be t h e u n i t

93

vector

in

j > i.

co .

for

eij

= ej

Then e a c h s e r i e s

Jim ~ t.e i.

i j=l

Set

J

J

Corollary

Ee.. j lj

i = lim E t.e.

II{tj}ll 8.4

i jffil J J

( 1. is

Much o f t h e

for

is C-convergent.

= = Z t.e.

j=l

1 ~ j ~ i

J J'

but the

and

eij For

convergence

= 0 {tj}

is

for E co ,

not uniform

Note a l s o t h a t the analogue o f the c o n c l u s i o n

false

for

material

this

particular

from this

section

example. is contained

in

[73].

in

10.

Imbedding

In this spaces

and

is

ization

~

result

to contain

then

on vector

cient

for

condition

As a c o r o l l a r y of

as a

The r e s u l t s

1.4 of Diestel Rosenthal's

are

([29]).

~x i

in

this

X

is

said

X

co .

for

each

in

The f o l l o w i n g

series

which will

Proposition

(i) (ii)

(iii)

I.

Zx i {Z x i leo

result

be needed

Let

x

for each

finite}

6 X

{t i} E c o ,

~

as

operators

on

~.

of section

a n d Uhl e m p l o y

a Banach space.

(such

A ser-

Cauchy

series

([21])).

(w.u.c.)

the

are

sometimes

Such series series

characterizations

The f o l l o w i n g

Ztix i

charto

to those

consider

several

is

to

isomorphic

are

is w.u.c. : o c ~

on

a suffi-

isomorphic

Ze i

(norm)

bounded

converges

may in

of w.u.c.

later.

{x i } ~ X.

a

!

example,

gives

c o,

Lemma.

denote

convergent

X; f o r

to

of Pelczynski

t o b e "~v__eakly u n c o n d i t i o n a l l y

weakly unconditionally

not be convergent

similar

Matrix

m

~ I< x , x i >I < ~

called

very

will

I

if

a subspace

a

general

the Diestel-Faires

Whereas Diestel

section

This

for

character-

isomorphic

on b o u n d e d l i n e a r

lemma, we u s e t h e B a s i c

Throughout ies

cct~tainin~

of proof

a n d Uhl

co .

a subspace

o f T h e o r e m 7 we o b t a i n

and style

to

I n T h e o r e m 7 we g i v e

to contain

of Rosenthal

condition

and a result

operators.

B-spaces

result

subspaces

sequence

I n T h e o r e m 3 we

a sufficient

isomorphic

measures

a B-space

X.

the Bessaga-Pelczynski

which contain

converging

well

which gives

employed to give

of Diestel

acterization

Banach space

a subspace

unconditionally

g~.

imbedding the classical

in a given

of B-spaces

result

~

we c o n s i d e r

a general

Banach space result

and

section

co

establish

co

equivalent:

95 (iv)

sup{ ~ I < x , x i > I : I l x ' l l i=l

~

1}

and

-- M < -

CO

IliZ__ltiXill

~< M l l ( t i l I I

for each

I t i} E c o .

n

U ~- t~x~U ~ M | ( t i } D

(v)

i=l

Proof:

for each

Assume ( i )

J

and l e t

m

i

x

be t h e f i n i t e

t

£X

and,

o 6 J.

thus,

Hence,

norm b o u n d e d .

Assume ( i i ) .

Zxi

If

n 6 1~.

Let

That

M> 0

(ii)

~.

Then f o r

, xi > 1

{ Z x i : o 6 J} i6o

is,

of

i

> I ~Z { <x i=l

the set

is weakly bounded

holds.

be s u c h t h a t

,

o 6 J.

and

subsets

co

,{ < x ,

i60

for each

{t i} 6 c o

" *

}I Z xi} I ~ M i6o

for

t

{Ix {{ ~ i ,

{ ~ < x ,x i > { ~ M

for

o 6 J

so that

i6o I < x , x i > ] ~ 2M ( [ 5 6 ]

1.1.2).

If

{t i} 6 c o

and

i6o n

n > m,

I{ [

n

i=m

2M sup{Itil

tix ill

II

(I)

For

x t . x . II i=l , I

,

i=m

: m ~ i ~ n}

Assume ( i i i ) .

,

= sup{ I Z t i < x , x i > } : and

X

i=l

{{ '~ 1} o

(iii) holds.

{t i} E c o ,

= sup{

{Ix

ti <

we h a v e

x

,xi>

: IIx

11 ~ 1) < t

Thus, in

for each

£I,

and ( i )

x

6 X , implies

[Ix that

l I ( I,

the sequence

{< x , x i > }

is

the set

0

B = {{<x , x i > ) is weak~-bounded in

~I.

Hence,

E ~I B

: {Ix II ( I} i s norm b o u n d e d i n

Z1,

i.e.,

96 w

0

sup( ~" I<x ,xi>l

: IIx II ~

i=l

For

{t i} £ c o ,

(1)

implies that

1}

= M <-.

{} ~ t . x . } { i=l i i

~ M {{{til}{

or (iv)

holds. That (iv)

implies

Thus,

- (iv)

suffices

(i)

linear

extension, M.

Coo

T{ti)

bounded l i n e a r

(v) implies

(v) holds.

subspace

s e q u e n c e s by

of

still

co

on

(iv)

.

T,

implies

(v),

it

(iii).

consisting

Coo

d e n o t e d by

if

and s i n c e

Define the linear

= i-EltiXi

operator

Thus,

and ( v ) i s c l e a r .

are equivalent

t o show t h a t

Assume t h a t

to

(i)

operator

of the finitely

Then ( v ) i m p l i e s and, to

{t i} 6 c o , we h a v e

therefore, co

T

that

non-zero T

is a

h a s a bounded l i n e a r

w i t h norm l e s s

T{t i} =

on t h e

Z tix i, i=l

than or equal and ( i i i )

holds. These properties

of w.u.c,

them a r e g i v e n in [21] We n e x t d e r i v e obtain

series

a r e w e l l - k n o w n and most o f

5.2.

the basic matrix result

t h e main r e s u l t s

of this

section.

which w i l l This result

be e m p l o y e d t o i s b a s e d on t h e

b a s i c m a t r i x Lemma 2.1 and i s a s i m p l e c o n s e q u e n c e o f Lemma 2 . 1 .

Lemma 2. li.m x i j J {mi}

= O

xij

Pick

6 X

for each

such that

Proof: eij

Let

eij

= e/2i+j+l).

~ i=l

be s u c h t h a t i.

Given

limi x i j

e > 0

= 0

there

for each

exists

j

and

a subsequence

)- I I II < e. j#i Xmimj j=l

> 0

such that

Let

(m i)

)- eij < e i,j

(for example,

be the subsequence given by the matrix

97 Lena

2.1

(applied

to

[]xij[[).

Then

[[Xm m ][ < e i j

for

i # j

ij and t h e r e s u l t

follows

easily.

U s i n g Lemma 2 we now e s t a b l i s h imbedding of

T h e o r e m 3.

in

llxi]l {mi }

X

) 6 > 0

such that

{mi}

T { t i} = ~ t i x n i=l i

into

X.

Proof:

concerning

the

X.

Suppose that

such that quence

co

our main result

contains for each

i.

X

{n i}

a topological

by t h e c l o s e d

series

Then t h e r e

for any subsequence defines

By r e p l a c i n g

a w.u.c,

Ex i

which is

exists

a subse-

of

isomorphism

T

of

subspace generated-by

co

the g

{xi},

we may a s s u m e t h a t

X

is separable.

,

such that

For each

i

pick

I

xi 6 X

J

[[xi[ [ = 1

and

<xi,xi > = [[xil I.

By the Banach-AIaoglu

I

Theorem ~

x

{x i}

has a subsequence

which converges

weak ~

t o an e l e m e n t

0

t

6 X .

To a v o i d c u m b e r s o m e n o t a t i o n I

weak ~. since

Now

lim <x ,xi> = O.

a

xi * x

) 5 -

[<x , x i > [

) 5/2

Again to avoid cumbersome

for

large

notation,

i assume

0

[<x i - x , x i > [ The m a t r i x

rix Lena

assume that

!

[<x i - x , x i > [

0

that

l

later,

2.

> 6/2

for

[<x i - x , x j > ]

Let

o f lemma 2 w i t h

{mi}

all

i.

satisfies

the condition

be t h e s u b s e q u e n c e

o f ~he M a t -

g i v e n by t h e c o n c l u s i o n

e = 6/4.

Now d e f i n e

a bounded linear

operator

T : co * X

by

cO

T ( t i } =,~:ltixmi..=

(Note this |

by Proposition o f Lemma 2,

I. )

If

map i s w e l l - d e f i n e d J

and i s c o n t i n u o u s

0

z i -- Xmi - x ,

then, using the conclusion

98 211T[ti}lt )

)

ItilS/2

I < z i, T[tj} > I )

- II(tj}[]5/4.

211T{tj}ll

)

It i l. - j # ~ i l t j l

T a k i n g t h e supremum o v e r

ll{tj}ll6/4

which i m p l i e s t h a t

T

i

l l gives

has a bounded

inverse. The same c o m p u t a t i o n h o l d s f o r any s u b s e q u e n c e This result

{n i}

of

(m i } -

was d e r i v e d i n [69] by u s i n g a form o f t h e A n t o s i k -

M i k u s i n s k i D i a g o n a l Theorem. ( S e e a l s o

[13].)

The p r o o f g i v e n h e r e i s

somewhat s i m p l e r and i s b a s e d on Lemma 2 which i s a more e l e m e n t a r y result

t h a n t h e D i a g o n a l Theorem.

We now g i v e s e v e r a l a classic

result

Corollary

4.

is subseries phic to

Proof:

First

o f B e s s a g a and P e l c z y n s k i on w . u . c ,

The B - s p a c e

X

X

contains

we d e r i v e

series

i s such t h a t e v e r y w . u . c ,

c o n v e r g e n t (norm) i f f

Suppose

X

contains

convergent.

not c o n v e r g e . in

o f Theorem 3.

[[21]).

series

in

X

no s u b s p a c e i s o m o r -

co .

subseries

{pj}

applications

~

and s a t i s f i e s

Then t h e r e

Hence, t h e r e such that

Pj+l z j = i=pj+lxniZ .

a series

is a

IIzjll

for each

1 (ii),

is obvious since

but not s u b s e r i e s

B-spaces ([21]);

l a r y 4 b a s e d on a r e s u l t

sequence

j , where

the series

convergent

B e s s a g a and P e t c z y n s k i d e r i v e C o r o l l a r y s e q u e n c e s in

which does

Zzj

is w.u.c.

By Theorem 3,

X contains

co .

The o t h e r i m p l i c a t i o n which i s w . u . c ,

HXni

but not

and an i n c r e a s i n g

t h e h y p o t h e s i s o f Theorem 3.

a subspace isomorphic to

which i s w . u . c ,

is a subseries 5 > 0

> 5

By P r o p o s i t i o n

~x i

Diestel

of Rosenthal

co

contains

a series

[namely, Zej). 4 from r e s u l t s

on b a s i c

and Uhl g i v e a p r o o f o f C o r o l ([34]

1.4.5).

The methods

employed in t h e p r o o f g i v e n above s h o u l d be c o n t r a s t e d

with those of

99 Diestel

a n d Uhl

We n e x t

([341)

derive

Theorem 3 ([34]

Corollary

S.

m : ~ ÷ X strongly

Proof: {A i }

A

in

m ~

of Diestel

b e an a l g e b r a

a bounded,

additive,

If

a result

lemma i s

on v e c t o r

used.

measures

from

1.4.2).

Let

is

where the Rosenthal

finitely

then

is

not

and a

X

of subsets additive

contains

strongly 6 > 0

of a set

set

function

a subspace

additive,

there

such that

S.

which is

isomorphic

is

]lm(Ai)[[

If

to

a disjoint > 5 for

not

co .

sequence

each

i.

0

(See chapters

1 or 4.)

For

x

E X ,

the

~

scalar

set

function

l

x m

is

bounded and finitely

additive

and,

therefore,

has bounded

w

variation.

Hence,

is w.u.c.

T h e o r e m 3 now g i v e s

A proof is

E [<x , m ( A j ) > [

of Corollary

b a s e d on R o s e n t h a l ' s Finally

ing operator

X

if series

ditionally

converging

([58]).

Corollary

Let

not unconditionally I 1 : co ÷ X

Y

and

Y

w.u.c,

series

a n d Uhl i n

Zm(Aj)

[34]

operator

operators

T : X ~ Y converging. 12 : c o ÷ Y

of Pelczynski

is

operator

an u n c o n d i t i o n a l l y

and

in

X

into

B-spaces on

CI

gives

converging result

the

on u n T

converg-

subseries

A weakly compact operator

and in certain

interesting

a result

series

([58]).

unconditionally

following

converging

6.

in

the

A bounded linear

a B-space

The i d e n t i t y

which is

We h a v e t h e tionally

by Diestel

operators.

carries

convergent

is,

Lemma.

into

T

That

result.

5 is given

converging

from a B-space

operator

the

we u s e T h e o r e m 3 t o d e r i v e

conditionally

true

< ~.

converse

is unconis

also

an e x a m p l e o f a n

but not weakly compact.

of Pelczynski

onmancondi-

([58]).

be a b o u n d e d l i n e a r Then t h e r e such that

exist

operator

which is

isomorphisms

TI 1 = 12 ( i . e . ,

T

has a

100

bounded i n v e r s e

Proof: that

By h y p o t h e s i s ZTx i

contains that

Z-Txi

{pi }

Since

[[x[]

w.u.c,

in

llujll

• 6 /

Y

such that

l[ T x [ l

/

Y,

Ilzj[I

IITII.

(m i}

derives

exist

6 • 0

z-'rx i

and

k'Tuj

is w.u.c.

x E X, t h e s e r i e s

~uj

is

and, m o r e o v e r ,

isomorphisms Evidently

of basic

such

and a s u b -

z j = Tuj

the series

such that

Corollary

X

we may a s w e l l assume

where

for each

in

Since

A p p l y i n g Theorem 3 t o t h e s e r i e s

respectively.

on t h e e x i s t e n c e

there

• 5,

1 (ii)

~x i

convergent.

1 (ii),

by P r o p o s i t i o n

Pelczynski

lished

Thus,

][T[[

is a subsequence

and

series

w h i c h i s not c o n v e r g e n t ,

By P r o p o s i t i o n

)

Co).

a w.u.c,

but not s u b s e r i e s

I 2 { t j} = ~ t j T u m j d e f i n e X

exists

i s not c o n v e r g e n t .

Pj+I ~ x i. i=pj+l

there

there

is w.u.c,

a subseries

sequence uj =

on a s u b s p a c e i s o m o r p h i c t o

Zuj

I i ( t j} = ZtjUmj I1

and

12

from

and

Z-Tuj

and co

into

TI 1 = 12.

6 by u s i n g a d e e p t h e o r e m o f

sequences

([58]).

Corollary

in [69] by e m p l o y i n g t h e A n t o s i k - M i k u s i n s k i

[21]

6 is estab-

D i a g o n a l Theorem.

The p r o o f a b o v e i s o f a much more e l e m e n t a r y c h a r a c t e r . The c o n v e r s e o f C o r o l l a r y

of unconditionally

characterization

We now c o n s i d e r In what f o l l o w s , J E N, £ = ( d )

6 h o l d s and f u r n i s h e s

will

converging operators

t h e p r o b l e m o f imbedding let

P

an i n t e r e s t i n g ([43]).

Z=

in

X.

be t h e power s e t o f

~.

If

denote the subspace of

t h o s e sequences which v a n i s h o u t s i d e

of

£= J.

which c o n s i s t s We e s t a b l i s h

of

the ana-

l o g u e o f Theorem 3 f o r m e a s u r e s .

Theorem 7. {~(j)}

Let

~ : ~ ÷ X be bounded and f i n i t e l y a d d i t i v e .

d o e s n ' t converge to

O,

If

then t h e r e e x i s t s a subsequence

101

{n i}

such that for any subsequence

defines a topological isomorphism

{pi } of

{ni},

T : ~=(J) ÷ X,

T~ = Ij~d~

where

J = {pj : j E ~}.

Proof:

We may assume ( b y p a s s i n g t o a s u b s e q u e n c e i f n e c e s s a r y )

there exists a ,

6 > 0

llplj)ll > 6 for each

such that

0

I

xj E X

such that

j.

that

Pick

I

IIxjll = 1

and

<xj,g(j)> = II~(J)II.

Let

Xo

be

xj

|

be t h e c l o s e d

s u b s p a c e s p a n n e d by

{g(j)

: j E N}.

Let

*

zj

0

restricted to the subspace

X o.

Since

Xo

is separable,

{zj}

is

w

O(Xo,X o)

relatively sequentially compact and, therefore, has a w

subsequence which i s

t

O(Xo,X o)

c o n v e r g e n t t o an e l e m e n t

!

z E Xo

I

with,

!

llz II ~ I.

O(Xo,X o}

For convenience, of notation,, assume that, {z~}

convergent

to

z .

Extend

z

t o an e l e m e n t

x

is

E X

I

with

Ilx

II

~ 1.

T h u s , we h a v e ,

(21

w

lim. <xj - x , g(i)> = 0 J

for each I

Consider t h e matrix

i.

I

[Yij] = [<x i - x , ~(j)>].

0

[Yii [ > 5 additive

,

[<x , g ( i ) > [

so,

> 5/2

for

large

i

since

again for convenience of notation, ]Yii ] > 6 / 2

(3)

Now

for a l l

x g

is strongly

assume t h a t

i. o

By (2) and t h e s t r o n g lim Yij = 0

additivity

for each

j

and

i

of each

(x i - x )g,

lim Yij = 0

for each

we h a v e i.

j

Therefore,

we may a p p l y Lemma 2 and o b t a i n

a subsequence

{mi }

such

that

(4)

~

i=l j#i ,

Since each

m

(x i - x )~

Lemma implies that w

lYmlmj I < 6 / 4 . ' '

{m i}

is strongly additive, Drewnowski's

has a subsequence

{n i}

such that each

0

(Xmi - x )~

is countably additive on the o-algebra ~enerated by

102

{nil.

Put

T : ~(J)

J = {n i : i E N). * X

by

operator

T$ = ~ j ~ d ~ .

From t h e c o u n t a b l e

(5)

D e f i n e a bounded l i n e a r

additivity,

(3) and ( 4 ) ,

2][T~]] * ]<Xni-X ,

] = [

we h a v e

d(Xni-X )~] =

[j=l ~ tnj<xni-x ,~(nj) > { ~ [tnl[. {Ynlnl { .. -

j$iZ[ t n j [ where

> [tnl[ 5 / 2

lYninj[

~ = {tj).

-

[[¢][5/4,

T a k i n g t h e supremum o v e r

[]T~[[ ~ ][~][6/8,

or

T

h o l d s f o r any s u b s e q u e n c e o f

is established

matrix type result.

(See also

in [13]

and F a i r e s

Corollary

8.

m : Z * X Then

X

Proof:

Let

Z

[72].)

If

m

by

finitely

is not strongly and a

o f Theorem 7 and t h e r e s u l t

~

spaces isomorphic to

i s bounded,

and l e t

additive.

are a disjoint I[m(Ej)[I

satisfies

~ 5.

Define

the conditions

is immediate.

characterization ~.

S

~.

such that Then

of a set

but not s t r o n g l y

there

the converse of Corollary

g i v e s an i n t e r e s t i n g

~(o) = E e i iEo

of subsets

additive,

5 ~ 0

result

[34]).

additive

g(A) = m( U E ~ ) . jEA J

Note that

the following

a subspace isomorphic to

{Ej} ~ E

: F * X

1.4.2,

be a o - a l g e b r a

be bounded,

contains

sequence

([33]

{ni}.

by means o f a n o t h e r

Theorem 7 h a s an i m m e d i a t e c o r o l l a r y of Diestel

i n (5) i m p l i e s

h a s a bounded i n v e r s e .

The same c a l c u l a t i o n This result

i

(The s e t finitely

8 also holds and,

therefore,

of B-spaces containing function

additive

~ : 2 * ~

sub-

defined

but not s t r o n g l y

by

additive.)

103

Corollary 1.4).

8 has a number o f i n t e r e s t i n g

We p r e s e n t

two s u c h a p p l i c a t i o n s .

applications

First

due t o B e s s a g a and P e l c z y n s k i ( [ 2 1 ] ) .

Corollary

If

X

contains

X

contains

no s u b s p a c e i s o m o r p h i c t o

no s u b s p a c e i s o m o r p h i c t o

[33]

we c o n s i d e r t h e f o l -

lowing r e s u l t

9.

(see

£~,

then

c o.

I

Proof:

Let

Zx i

be a w . u . c ,

series

in

X .

Then,

w

in p a r t i c u l a r , w

Zl<xi,x>l < -

for each

subseries

convergent.

function

g : P ÷ X

x 6 X.

Thus, t h e s e r i e s

Zx i

Now d e f i n e a bounded, f i n i t e l y by

g(o)

x I.

By C o r o l l a r y

i s weak ~ additive 8,

~

set

is

i strongly

additive.

Hence, f o r any s u b s e q u e n c e

{mi},

w

lim [ [ ~ ( m i ) [ [

= lim [[Xmil[ = O.

Theorem 7.1 i m p l i e s t h a t

the series

I

Ex i

i s norm s u b s e r i e s

convergent.

As a s e c o n d a p p l i c a t i o n , Faires

on weak ~ s u b s e r i e s

earlier

in s e c t i o n s

Corollary Zx i

I0.

Let

Proof:

we e s t a b l i s h

the result

convergent series

of Dies,el

w h i c h was r e f e r r e d

and

to

3 and 7.

X

i s weak ~ s u b s e r i e s

subseries

Corollary 4 gives the result.

c o n t a i n no s u b s p a c e i s o m o r p h i c t o c o n v e r g e n t in

X ,

then

Ex i

Z~.

If

i s norm

convergent.

D e f i n e a bounded, f i n i t e l y

additive

set function

~ : ~ ÷ X

!

by

g(o) =i~oxi.

By C o r o l l a r y

8

g

is strongly

additive.

Hence, f o r

in norm.

Theorem 7.1

)

any subsequence

{mi},

lim ~(m i) = lim Xmi = 0 w

implies that the series

~x i is norm subseries convergent.

The c o n v e r s e o f C o r o l l a r y characterization

i0 a l s o h o l d s and g i v e s an i n t e r e s t i n g

of dual spaces c o n t a i n i n g

copies of

~

(see

[33]

104

Corollary

1.2 f o r the p r o o f ) .

Finally, result

we o b t a i n as a c o r o I I a r y

of Rosenthal

Corollary If there

II.

i s an i n f i n i t e

restricted

Proof:

to

ll~(i)]l

A.

~(A) = T(CA), Then

By h y p o t h e s e s t h e r e

= I[TC{i}II ~ 6

Rosenthal actually

T

i s an i n f i n i t e

by

function of

11 in P r o p o s i t i o n

such t h a t

be bounded and l i n e a r . restricted

to

Co(I)

J ~ I

such t h a t

where

CA

T

i s an i s o m o r p h i s m .

g : P * X

Tf = f f d g .

T : £~ * X

I E N

then there

~(J)

Define

characteristic and

([65]).

(Rosenthal) Let

i s an i s o m o r p h i s m ,

o f Theorem 7 t h e f o l l o w i n g

for obtains

1.2 of [65].

which i s g i v e n in C o r o l l a r y

i 6 I.

g

denotes the

i s bounded, f i n i t e l y

is a

6 > 0

additive

such that

Thus, Theorem 7 g i v e s t h e r e s u l t .

a more g e n e r a l r e s u l t

than Corollary

For the c o u n t a b l e case of h i s r e s u l t

11 o u r methods a r e much s i m p l e r .

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11.

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33.

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34.

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36.

L. Drewnowski, E q u i v a l e n c e o f B r o o k s - J e w e t t , V a t a l i - H a n - S a k s , and Nikodym Theorems, B u l l . Acad. Polon. S c i . , 20 (1972}, 725-731.

37.

L. Drewnowski, Uniform Boundedness P r i n c i p l e f o r F i n i - t e l y A d d i t i v e V e c t o r Measures, B u l l . Acad. Polon. S c i . , 21 (1973), 115-118.

L i n e a r Spaces, S p r i n g e r - V e r l a g , B e r l i n ,

1973.

Measures, Amer. Math. Soc. S u r v e y s

o f FJ~nctional A n a l y s i s , N o r t h - H o l l a n d ,

107

38.

N. Dunford and J. Schwartz, Linear Overators I, Interscience, N. Y., 1958.

39.

H. Hahn, Uber Folgen l i n e a r e n O p e r a t i o n e n , Monatsch. Fur Math. und P h y s . , 32 (1922), 1-88.

40.

R. Haydon, A n o n - r e f l e x i v e Grothendieck space t h a t does not c o n t a i n £~, I s r a e l d. M., 40 (1981), 65-73.

41.

E. Hewitt and K. Stromberg,

42.

T. H. Hildebrandt, On Uniform Limitedness of Sets of Functional Operations, Bull. Amer. Math. Soc., 29 (1923), 309-315.

43.

J. Howard, The Comparison o f an U n c o n d i t i o n a l l y Conve:'ging O p e r a t o r , S t u d i a Math., 33 (1969), 295-298.

44.

N. J. Kalton, Spaces of Compact Operators, Math. Ann., 208 (1974), 267-278.

45.

C. Klis, An Example of Noncomplete Normed (K)-Space, Bull. Acad.

Real and Abstract Analysis, Springer-Verlag, N. Y., 1965.

Polon. Sci., 26 (1978), 415-420. 46.

G. Kothe, Topological Yector Spaces I, S p r i n g e r - V e r l a g , N. Y., 1969.

47.

G, Kothe, Tovological Vector Svaces II, Springer-Verlag, N. Y., 1979.

48.

I. Labuda and Z. L i p e c k i , On S u b s e r i e s Convergent S e r i e s and m-Quasi Bases in Topological L i n e a r Spaces, Manuscripta Math., 38 (1982), 87-98.

49.

J . L i n d e n s t r a u s s and L. T z a f r i r i , Springer-Verlag, N. Y., 1977.

50.

I. Maddox, Elements o f F u n c t i o n a l A n a l y s i s , Cambridge Univ. P r e s s , 1970.

51.

S. Mazur and W. O r l i c z , Uber Folgen l i n e a r e n O p e r a t i o n e n , S t u d i a Math., 4 (1933), 152-157.

52.

S. Mazur and W. O r l i c z , Sur l e s espaces m e t r i q u e s l i n e a i r e s S t u d i a Math., 13 (1953), 137-179.

53.

J. M i k u s i n s k i , A Theorem on Vector M a t r i c e s and i t s A p p l i c a t i o n s in Measure Theory and F u n c t i o n a l A n a l y s i s , B u l l . Acad. Polon. S c i . , 18 (1970), 193-196.

54.

J . M i k u s i n s k i , On a Theorem o f Nikodym on Bounded Measures, B u l l . Acad. Polon. S c i . , 19 (1971), 441-443.

55.

W. O r l i c z , B e i t r a g e zur Theorie der O r t h o g o n a l e n t wicklungen I I , S t u d i a Math., 1 (1929), 241-255.

56.

E. Pap, F u n k c i o n a l n a ~_naliza, I n s t i t u t e o f Mathematics, Novi Sad, 1982.

C l a s s i c a l Banach S p a ~

I,

If,

108

57.

E. Pap, Contributions to Functional Analysis on Convergence Spaces, General Topology and its Relations to Modern Analysis and Algebra V, Prague, 1981.

58.

A. Pelczynski, On Strictly Singular and Strictly Cosingular Operators, Bull. Acad. Polon. Sci., 13 (1965), 31-36.

59.

B. J. Perils, On Integration in Vector Spaces, Trans Amer. Math. Soc., 49 (1938), 277-304.

60.

R. Phillips, On Linear Transformations, Trans. Amer. Math. Soc., 48 (1940), 516-541.

61.

A Pietsch, Nuklesro Lokalkonvex¢ Raum~, Akademic-Verlag, Berlin, 1965.

62.

A. P. Robertson, Unconditional Convergence and the Vitali-HahnSaks Theorem, Bull. Soc. Math., France, Suppl. Mem., 31-32 (1972), 335-341.

63.

S. Rolewicz, Metric Linear Spaces, P o l i s h Scien. P u b l . , Warsaw, 1972.

64.

H. Rosenthal, On Complemented and Quasi-complemented Subspaces of Quotients of C(S) for Stonian S, Proc. Nat. Acad. Sci., U.S.A., 59 (1968), 361-364.

65.

H. Rosenthal, On Relatively Disjoint Families of Measures with Some Applications to Banach Space Theory, Studia Math., 37 (1970), 13-36.

66.

S. Saxon, Some normed barrelled spaces which are not Baire, Math. Ann., 209 (1974), 153-160.

67.

W. Schachermayer, On Some classical Measure-Theoretic Theorems for non-sigma-complete Boolean Algebras, Dissert. Math., to appear.

68.

J. Sember, On Summing Sequences of Math., 11 (1981), 419-425.

69.

C. Swartz, Applications of the Mikusinski Diagonal Theorem, Bull. Acad. Polon. Sci., 26 (1978), 421-424.

70.

C. Swartz, A Generalized Orlicz-Pettis Theorem and Applications, Math. Zeit., 163 (1978), 283-290.

71.

C. Swartz, Orlicz-Pettis Topologies in Function Spaces, Publ. de L'Inst. Math., 26 (1979), 289-292.

72.

C. Swartz, A Lemma of Labuda as a Diagonal Theorem, Bull. Acad. Polon. Sci., 30 {1982), 493-497.

73.

C. Swartz, The Schur Lemma for Bounded Multiplier Convergent Series, Math. Ann., 263 (1983), 283-288.

O's

and

1"s, Rocky Mr. J.

109

74.

C. Swartz, An Abstract Orlicz-Pettis Theorem, Bull. Acad. Polon. Sci., to appear.

75

C. Swartz, Continuity and Hypocontinuity for Bilinear Maps, Math. Zeit., 186 {1984), 321-329.

76

C. Swartz, The Schur and Hahn Theorems for Operator Matrices, Rocky Mountain d. Math., to appear.

77

G . E . F . Thomas, L'integration par rapport a une mesure de Radon vectorielle, Ann. Inst. Fourier, 20 (1970), 55-191.

78

I. Tweddle, U n c o n d i t i o n a l Convergence and Vector-Valued Measures, d. London Math. S o c . , 2 {1970), 603-610.

79

A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N. Y., 1978.

80

K. Yosida, F u n c t i o n a l An,a!ysis, S p r i n g e r - V e r l a g , N.Y., 1966.

INDEX

Absolute

Continuity

Uniform

.

.

.

.

.

.

Antosik-Mikusinski Basic

Matrix

Bounded

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

35

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

35

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

Theorem

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

7

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

98,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

45

. . . . . . . . . . . . . .

77

.

.

.

.

.

multiplier .

.

.

.

.

Bourbaki

.

.

Bessaga-Pelczynski Biadditive

.

.

.

.

.

.

convergent

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

50,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

32

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

92

Diagonal

Theorem

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

Diestel

.

99

Brooks-Jewett C-convergent

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

6

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

37

Diestel-Faires Drewnowski's (E)

Lena

(property)

.

.

103

51

102,103

Equicontinuity right

(left),

separate

Equihypocontinuity Exhaustive

.

.

.

.

.

.

. .

General

Uniform

General

Banach-Steinhaus

Hahn

.

.

.

.

.

. .

. . . . . . . . . . . . . .

. .

. .

Boundedness

.

.

Hypocontinuity

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Principle

Theorem .

.

.

:~

.

53 56 6,

. . . . . . . . . .

22

. . . . . . . . . . . . .

30

.

.

.

.

.

.

.

.

.

.

.

.

.

.

40,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

45

-bounded

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

13

X -compact

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

31

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

11

X -convergent -matrix -space

.

Mazur-Orlicz

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

9,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

12

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

47

32

80

II

111

Nikodym Boundedness Theorem

. . . . . . . . . . . . . . .

26, 28

Nikodym C o n v e r g e n c e Theorem

. . . . . . . . . . . . . . .

34

Normed g r o u p . . . . . . . . . . . . . . . . . . . . . . .

5

Operator

norm

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

5

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

18,

Orlicz-Pettis Phillips

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

81

Pelczynski

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

99

Quasi-norm

. . . . . . . . . . . . . . . . . . . . . . . .

Quasi-o-algebra Regular Rosenthal Schur

5

. . . . . . . . . . . . . . . . . . . . .

37

. . . . . . . . . . . . . . . . . . . . . . . . .

43

. . . . . . . . . . . . . . . . . . . . . . . .

104

. . . . . . . . . . . . . . . . . . . . . . . . . .

59

41, 77

Series Bounded multiplier convergent C-convergent

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

77 92

Subseries convergent

. . . . . . . . . . . . . . . .

58

Unordered convergent

. . . . . . . . . . . . . . . .

79

Weakly unconditionally Cauchy

. . . . . . . . . . . .

Weakly unconditionally convergent Strongly additive Uniformly

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

S t r o n g l y bounded

. . . . . . . . . . . . . . . . . . . . .

Subsequential Completeness Property Subseries convergent

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

94 94 6, 32 32 6, 32 37 58

Theorem Antosik-Mikusinski Banach-Steinhaus Basic ~ a t r i x

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

Bessaga-Pelczynski Bourbaki

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

Brooks-Jewett

. . . . . . . . . . . . . . . . . . . .

1 29, 30 7 98. 51 32

103

112

Diagonal

. . . . . . . . . . . . . . . . . . . . . .

D i e s t e l - F a i r e s

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1 .

.

102,

Diestel . . . . . . . . . . . . . . . . . . . . . . .

99

General

Uniform

. . . . . . . . . . . . .

22

General

Banach-Steinhaus

. . . . . . . . . . . . . .

30

Hahn

Boundednes$

. . . . . . . . . . . . . . . . . . . . . . . .

Kalton

40,

. . . . . . . . . . . . . . . . . . . . . . .

Mazur-Orlicz Nikodym

47

Boundedness

. . . . . . . . . . . . . . . . .

26,

Nikodym Convergence

. . . . . . . . . . . . . . . . .

34

. . . . . . . . . . . . . . . . . . . .

Pelczynski Rosenthal Schur

.

Toeplitz Uniform

. . . . . . . . . . . . . . . . . . . . .

99

. . . . . . . . . . . . . . . . . . . . . .

104

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

41, .

. . . . . . . . . . . . . . . . . . . . . . Boundedness

Vitali-Hahn-Saks Unconditionally absolute

Uniform

Boundedness countably

Uniformly

strongly Property

65, 67,

70

43

. . . . . . . . . . . . . . . . . . .

35

operator

Principle additive additive

. . . . . . . . . . .

. . . . . . . . . . . . . . . (UBP)

. . . . . . . . . . .

99 35 21

. . . . . . . . . . . . . . .

34

. . . . . . . . . . . . . . .

32

. . . . . . . . . . . . . . . . . . . . .

Weakly

unconditionally

Cauchy

(w.u.c.)

Weakly

unconditionally

convergent

59

80

21

continuity

Uniformly

.

28

77,

. . . . . . . . . . . . . . . . .

converging

Uniform

Urysohn

18,

. . . . . . . . . . . . . . . . . . . . . . . .

Thomas

80

63

. . . . . . . . . . . . . . . . . . . .

Orlicz-Pettis

103

12

. . . . . . . . . .

94

. . . . . . . . . . . .

94

NOTATION E,

F,

G (normed

groups)

[ [ (quasi-norm)

[[ (norm)

.

.

.

.

.

.

.

.

.

linear

spaces

.

5

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

o(X,Y)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

5

5

. . . . . . . . .

L(X,Y)

.

.

. . . . . . . . . . . . . . .

X, Y, Z (metric [[

.

5 5 .

.

5

Coo

. . . . . . . . . . . . . . . . . . . . . .

ej

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

12

mo

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

14

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

15

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

15

T(F)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

22

F-K

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

22

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

24

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

24

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

26,

S(Z) CE

ba

.

ca(E)

<<

. . . . . . . . . . . . . . . . . . . . . .

(~,c) c

12

.

.

(mo,C) (c,c)

35

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

38

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

39

. . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

43

b(x,')

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

45

o(E,F)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

45

~(E,F)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

46

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

46

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

48

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

48

~(E)

.

39

.

~(E,F)

K~(E)

.

[(F)

. . . . . . . . . . . . . . . . . . . . .

49

56

114

K(X,Y) CG(S)

. . . . . . . . . . . . . . . . . . . .

62

. . . . . . . . . . . . . . . . . . . . .

64

. . . . . . . . . . . . . . . . . . . .

68

B(S,Z) £P(G) ~F

. . . . . . . . . . . . . . . . . . . . .

69

. . . . . . . . . . . . . . . . . . . . . .

72

~(E,F) c(X)

. . . . . . . . . . . . . . . . . . . .

72

. . . . . . . . . . . . . . . . . . . . .

80

(~,c(X))

. . . . . . . . . . . . . . . . . . .

80

(mo,c(X))

. . . . . . . . . . . . . . . . . . .

80

£~(d)

. . . . . . . . . . . . . . . . . . . . .

i00