A Banach space with basis constant > 1

A Banach space with basis constant > 1. :PEg ENI~LO University of Stockholm and University of California at Berkeley

n ~e il IfaBanachspacehasaSchauderbasis fl, t h e n fiK ~-~ SUp~,~ll~i=l i h/HX /[ exists, where x = ~i~=la~d. Inf/?K t a k e n over all fi is called the basis c o n s t a n t of the B a n a c h space. I t is obvious t h a t if the B a n a c h space B has the basis c o n s t a n t p, t h e n e v e r y finite-dimensional snbspace C of B can be a p p r o x i m a t e d b y subspaces D , of B - b y a p p r o x i m a t i n g a set of basis vectors of C w i t h vectors of finite expansions in some basis - such t h a t each Dn can be e m b e d d e d into a finitedimensional subspace E~ of B, onto which there is a p r o j e c t i o n f r o m B of n o r m a r b i t r a r i l y close to p. I n this p a p e r we c o n s t r u c t a separable infinite-dimensionM B a n a e h space B with a two-dimensional subspace C 1 with the following properties: T h e r e is a p > 1 such t h a t , if D is a two-dimensional subspace of B sufficiently close to C 1 a n d E is a finite-dimensional subspaee of B containing D, t h e n t h e r e is no p r o j e c t i o n f r o m B onto E of n o r m ~ p. Thus the basis c o n s t a n t o f this 13anach space is > p. This seems to be b y now the strongest result in negative direction on the well-known basis problem. T h e p r e v i o u s l y strongest result seems to be Gurarii's e x a m p l e of a B a n a c h space where fiK > 1 for e v e r y ft. (See Singer [1] pp. 218--42.) W e now s t a r t b y giving a general and s o m e w h a t u npreeise description o f the ideas b e h i n d the c o n s t r u c t i o n and of the problems we meet. W e consider a twodimensional subspace C 1 of l~(F), where F is the set of pairs of positive integers. W e assume t h a t the p r o j e c t i o n c o n s t a n t of C 1 is > 1. Now our first a m b i t i o n will be to e m b e d C1 in a larger space El, such t h a t t h e r e is no p r o j e c t i o n of n o r m close to 1 f r o m E I onto spaces close to C 1 a n d such t h a t no subspaee C 2 of E1 containing a subspace of E1 sufficiently close to C 1 has a p r o j e c t i o n c o n s t a n t n e a r to 1. H o w e v e r , if we t r y to do this we h a v e to get control of quite m a n y linear spaces. I n order to describe h o w we o b t a i n the necessary simplifications, we give n o w a description o f the w a y we e s t i m a t e n o r m s of projections.

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L e t us a s s u m e t h a t C 1 is a t w o - d i m e n s i o n a l s u b s p a c e of a B a n a c h space B a n d t h a t 0, al, a2, . . . , a~ are n d- 1 points in C 1 such t h a t Ilajll = 1; j ~ 1, 2 , . . . , n, lla~--ajll <~1; i , j = 1 , 2 , . . . , n . W e s a y t h a t b E B is a central point for t h e I t is obvious t h a t d- 1 points, if Ilbll- -~ a n d lib - - ajll = g, 1. j = 1 , 2 , . . . , n . if b is t h e o n l y c e n t r a l p o i n t in B for t h e n + 1 points 0, a 1,a 2. . . . , a s a n d points f a r a w a y f r o m b are f a r f r o m being central points, t h e n a s u b s p a c e of B which contains a s u b s p a c e close to C 1 a n d o n t o which t h e r e is a p r o j e c t i o n f r o m B of n o r m close t o 1 m u s t c o n t a i n a v e c t o r n e a r to b. F o r otherwise b y such a p r o j e c t i o n b w o u l d be m a p p e d o n t o a p o i n t whose d i s t a n c e to s o m e of the points 0, 61, a 2. . . . , a , w o u l d be essentially larger t h a n }. This d e s c r i p t i o n suggests h o w we can get control of t h e p r o j e c t i o n c o n s t a n t s of s u b s p a c e s of B c o n t a i n i n g a s u b s p a c e close to C 1. I t n a m e l y gives t h a t we can restrict our a t t e n t i o n to s u b s p a c e s of B containing a s u b s p a c e close to C 1 a n d cont a i n i n g a v e c t o r close to b. I n our e x a m p l e we h a v e in C 1 t w o sets 0, all , a12. . . . ,61, , a n d 0, a21, 6 2 2 , . . . , a2, with u n i q u e central points bl a n d b2 a n d so we can restrict our a t t e n t i o n to s u b s p a c e s of B c o n t a i n i n g v e c t o r s close to bI a n d b2. T h e w a y in which we will build u p our B a n a c h space is as follows. W e s t a r t w i t h a s u b s p a c e C 1 of l~(F), g e n e r a t e d b y t w o v e c t o r s e1 a n d e 2. I n C 1 we h a v e t w o p o i n t sets w i t h u n i q u e c e n t r a l p o i n t s 89 a a n d 89 4 in B. T h e c o n s t r u c t i o n is m a d e so t h a t for e v e r y t w o - d i m e n s i o n a l s u b s p a c e D of lop(F) sufficiently close to C 1 t h e r e are p o i n t sets of D such t h a t 89 a a n d le4 are t h e u n i q u e central points in B for these p o i n t sets a n d points far f r o m 89 ~ a n d 89 4 are far f r o m being c e n t r a l points for t h e s e p o i n t sets. This is i m p o r t a n t since it gives t h a t t h e r e will be no a c c u m u l a t i o n of a p p r o x i m a t i o n s in our construction. N o w in e v e r y s u b s p a c e of B s u f f i c i e n t l y close to t h a t g e n e r a t e d b y e a a n d e4 t h e r e are p o i n t sets w i t h the u n i q u e c e n t r a l p o i n t s 89 5 a n d 89 G. T h e f a c t s j u s t described are p a r t o f L e m m a 2 below a n d give a n idea of h o w the discussion goes on. W e will f i n a l l y conclude t h a t if a s u b s p a c e E of B contains a s u b s p a c e close to C 1 a n d t h e r e is a p r o j e c t i o n f r o m B onto E of n o r m close to 1, t h e n E can n o t be f i n i t e - d i m e n s i o n a l . W e n o w define t h e v e c t o r s eY, eJ C l~(I), which g e n e r a t e B. W e let e~,~J d e n o t e the c o m p o n e n t of eJ which c o r r e s p o n d s to t h e p a i r (r, k). W e choose e~,k = 0.611 - - (k - - 1)- 0.0ol, k = 1, 2 , . . . , 12. W e t h e n choose 2 2 e~,l = - - 0.60o a n d choose e~,k, k = 2, 3, . .., 12 i n d u c t i v e l y b y t h e e q u a t i o n s 0.61l - - (k -- 1) 9 0.001 - - ( 0 , 9 9 9 5 ~ - 0 . 0 1 ( ] c - - 1))e~,k = 0 . 6 1 1 - - ( k - - 2 ) * 0.OOl - - ( 0 . 9 9 9 5 -10.Ol. ( / c - 1))e~,k_l. B y this choice we o b t a i n t h a t e~,k k - 1, 2 , . . . , 12 lies b e t w e e n - - 0.6oo a n d - - 0.6~2 a n d t h a t the c o m p o n e n t (1, k) is the largest of t h e 12 c o m p o n e n t s (1, 1), (1,2) . . . . , ( 1 , 1 2 ) of t h e v e c t o r e1 - (1 + 0 . o 1 . ( k - - 1))e 2, /c = 1, 2 . . . . . 12. W e also see t h a t if v~ a n d v 2 are s u f f i c i e n t l y close to e~ a n d e 2 in l~(F) t h e n t h e c o m p o n e n t (1, k) will be t h e largest of t h e 12 c o m p o n e n t s (1,1), ( 1 , 2 ) , . . . , ( 1 , 1 2 ) of v 1 - (1 + 0 . 0 ~ . ( k - - 1))v 2, k = 1 , 2 , . . . , 1 2 . We put el, k = e~.k = 0 if k > 12. C o m p o n e n t s (r, k) w i t h r > i will be d e f i n e d later. W e will h a v e need for such c o m p o n e n t s to s a v e the uniqueness of central points.

A BANACH

SPACE

WITtt

BASIS

CONSTANT

>

105

1

W e will n o w d e f i n e ea a n d e4 so t h a t 89 a is t h e u n i q u e c e n t r a l p o i n t in B for t h e 7 p o i n t s 0, (e 1 - - (1 d- 0.o1(k - - 1))e2)/lle 1 - - (1 q- 0.ol(k - - 1))e211, k = 1, 2 . . . . , 6 a n d so t h a t 89 4 is t h e u n i q u e c e n t r a l p o i n t in B for t h e 7 p o i n t s 0, (ea - - (1 q- 0.o1(k - - 1))e2)/lle 1 - - (1 q- 0.ol(k - - 1))e2][, k = 7, 8 . . . . . 12 a n d so t h a t this h o l d s e v e n if e 1 a n d e 2 are r e p l a c e d b y a p p r o x i m a t i n g v e c t o r s v 1 a n d v a . . = e 1,6 3 = 1 and we choose in B. T o this e n d we p u t e a3 , 1 = e 13, 2 = . e 31,7, e 31,8, 9 ~ 9 , {31,12 a in s u c h a w a y t h a t : (a) t h e d i s t a n c e in / ~ - n o r m b e t w e e n t h e 6 - v e c t o r 89 (%7, a el,8, 3 9 9 -, e 31,12) a n d e a c h o f t h e s e v e n 6 - v e c t o r s w h i c h c o n s i s t o f t h e c o m p o n e n t s (1, 7), (1, 8 ) , . . . , (1, 12) o f t h e v e c t o r s 0, (e1 - - (1 q- 0.01(k - - 1))e2)/l[e 1 - - (1 q- 0.01(k - - 1))e2[I, k = l, 2 . . . . , 6, is s t r i c t l y less t h a n 1. 1 2 2 e3 e3 e3 (b) t h e f o u r 6 - v e c t o r s ( e ~ , 7 , e l1,8, 9 9 " , e1,12), ., s 2 ( 1 , 7, 1 , 8 ~ 9 " 1,12) (s s " a n d (1, 1, 1, 1, 1, 1) are l i n e a r l y i n d e p e n d e n t 9 W e n e e d (a) since 89 shall be a c e n t r a l p o i n t e v e n if w e a p p r o x i m a t e el a n d e2 b y v 1 a n d v 2 a n d (b) since 89 d e f i n e d b e l o w shall be a u n i q u e c e n t r a l p o i n t . 4 e43,2, 9 9 9 el.6 4 in s u c h a w a y t h a t : W e p u t e l4, 7 __ e41,8 ~ 9 9 9 _- -,; e. 4 1,12 a n d c h o o s e e1,i, (al) t h e d i s t a n c e in / ~ - n o r m b e t w e e n t h e 6 - v e c t o r 89 9 (e~,l, e43,2, 9 . . , s 1,6) a n d e a c h o f t h e s e v e n 6 - v e e t o r s w h i c h consist o f t h e c o m p o n e n t s (1, 1), (1, 2), . . . , (1, 6) o f t h e v e c t o r s 0, (s _ (1 q- 0.01(k - - 1))e2)/[[e I - - (1 q- 0.01(k - - 1))e2[[, k = 7, 8 , . . . , 12, is s t r i c t l y less t h a n 89 (51) t h e f o u r 6 - v e c t o r s (e{,3, e{,2. . . . , s j = 1, 2, 4 a n d (1, 1, 1, 1, 1, 1) are linearly independent. A s a b o v e w e n e e d (al) since 1s shall be a c e n t r a l p o i n t e v e n if w e a p p r o x i m a t e e 1 a n d e2 b y v I a n d v e a n d ( b l ) s i n c e 89 shall be a u n i q u e c e n t r a l p o i n t . l~or k > 1 w e n o w c h o o s e s 3 1 a n d e41 , 1 2 ~ - k ~ s 2 W e see t h a t t h i s ~ s does n o t d e s t r o y t h a t ~ea a n d 89 4 a r e c e n t r a l p o i n t s f o r t h e sets d e s c r i b e d a b o v e 9 W e also o b t a i n t h a t t h e c o m p o n e n t s (1, 1), (1, 2 ) , . . . , (1, 12) of e3 - - (1 @ 0.01(k - - 1)) 9 e4, k = 1, 2 . . . . , 12 are all < 0.25 a n d t h a t t h e c o m p o n e n t (1, 12 @ k) is t h e l a r g e s t o f t h e c o m p o n e n t s (1, 13), (1, 1 4 ) , . . . , (1, 24) o f e 3 - - ( 1 - ~ 0 . e l ( k - - 1 ) ) . e ~, k = 1, 2 , . . . , 1 2 , a n d is > 1. T h e last s e n t e n c e is o b v i o u s l y t r u e e v e n if e 3 a n d e4 are r e p l a c e d b y a p p r o x i m a t i n g v e c t o r s v 3 a n d v4 in l ~ ( F ) . W e n o w c h o o s e e51,k,el,k6 k = 1, 2, . . . , 12, t o be all in t h e i n t e r v M (0.0s, 0.10) a n d c h o o s e t h e m in s u c h a w a y t h a t t h e six 6 - v e c t o r s (ei,1, s . . . . , e{,6), j = 1, 2 , . . . , 6, a r e l i n e a r l y i n d e p e n d e n t a n d s u c h t h a t t h e six 6 - v e c t o r s (e{,7, eJ,8 . . . . , e{,~2) are l i n e a r l y i n d e p e n d e n t . This is d o n e in o r d e r t o m a k e ~ea a n d 89 u n i q u e c e n t r a l p o i n t s . :For all k > ] we put s 1 , 1 2 + k ~-- s 1 , k and -"

-

6

",

-

4

s

~

s

k"

N o w for all k <- -1 2 j , j > -l- -, we p u t e 6+(2j-1) 0 a n d for all k >- - 1 a n d 3,k ~ j > 1 we p u t e6+(2i-1) 5 1,12j+k = el,k. F o r all k < 123", j >_ 1, we p u t e ~ 2 j = 0 a n d 6~2j 6 for all k > 1 a n d j > 1 w e p u t s = s k" N o w we h a v e t o i n t r o d u c e c o m p o n e n t s (r, k) w h e r e r > 1. F o r if we d i d n o t _

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do t h a t , we w o u l d o n l y h a v e o b t a i n e d t h e following: 89 2j+1 a n d 89 2i+2 are b o t h c e n t r a l p o i n t s for sets c o n s i s t i n g o f 0 a n d l i n e a r c o m b i n a t i o n s o f e 2j-1 a n d e 2i. This h o l d s e v e n if e 2j-~ a n d e 2j are a p p r o x i m a t e d b y v e c t o r s v 2J'-I a n d v 2j in B. B u t 89 2j+1 a n d 89 2i+2 will n o t be u n i q u e c e n t r a l p o i n t s . F o r c e r t a i n l i n e a r c o m b i n a t i o n s o f el:s, i ~ 2 j - - 2 or i > 2 j ~ 5 c o u l d be a d d e d t o 89 2j+1 a n d 89 2i+2 w i t h o u t d e s t r o y i n g t h e i r p r o p e r t y o f b e i n g c e n t r a l p o i n t s . 2j- 1 2j F o r all j >_ 1 a n d k > 1, e[(j+3)/2l, k j =- l, %'+2,k = 0.1 a n d e1+2,k ~ - - 0.1. 2j- 1 2j T h e d e f i n i t i o n o f ej+z,k a n d ey+2,k is m a d e so t h a t 89 2j+1 a n d 89 2j+2 are c e n t r a l p o i n t s e v e n if e 2~-1 a n d e2j are a p p r o x i m a t e d b y v e c t o r s v 2j-~ a n d v 2j in B. l % r t h o s e t r i p l e t s (j, r, k) w h e r e e~,k is n o t y e t d e f i n e d , we p u t e]]k ~ 0.1 9 ( - - 1) [(~-1/2)1. This last d e f i n i t i o n will s a v e t h e u n i q u e n e s s o f 89 2y+~ a n d 89 =j+2 w i t h r e s p e c t t o l i n e a r c o m b i n a t i o n s o f el:s, i < 2 j - 2 or i > 2 j ~ - 5 . F o r if we a d d s u c h a linear c o m b i n a t i o n t o 89 2j+~ or 89 2j+2 t h e d i s t a n c e t o 0 will be > 89 W e n o w r e s u m e in t h r e e l e m m a s t h e i m m e d i a t e c o n s e q u e n c e s o f o u r d e f i n i t i o n s o f t h e v e c t o r s e ~', j > 1. LEMMA 1. There exists an s > 0 such that i f I]v2j-1 -- e2j-lll < ~ and ]Iv2j -- e2J[] <_ e holds for two vectors v 2i-1 and v 2j in B, j >_ 1, then v2i i _ (1 ~ - 0 . 0 1 ( k - 1))v ~j has its largest component on the same place as e 2j-~ - - (1 ~- 0.01(k - - 1))e 2j, k : l, 2, . . . , 12, namely some component (r, m) with r = 1, and the distance between 89 2i+1 and each of the elements 0, (v2J 1 _ _ (1 -~- 0.01(]c - - 1))v2J)/Hv2j-1 -- (1 -~- 0.01(k - - 1))v2J[[, k = 1, 2 , . . . , 6, is 89

Obviously a similar result holds for 89 2j+2 for k --~ 7, 8 . . . . .

12.

LEMMA 2. Let v 2j-I and V 2j be as in L e m m a 1. only vector of B such that the distance between the vector and (v 2j-x - - (1 ~-~ O.01(k ]))v2J)/llv2j-1 - (1 -~- O.01(k - 1))v2Jl[, k and there are positive constants K and 81 such that i f g E B then the distance between 89 and some of the __

O, (V 2j 1 - - ( l ~ - 0.01(]~ - - 1))v2J)/]]V - - (1 ~ - 0.01(]~ - - ]))v2J[],

T h e n 89 2j+1 is the each of the vectors 0, : 1, 2 , . . . , 6, is ~2 and []g -- e2/+1]1 z 5, 1

]~ =

1, 2 , . . . ,

6,

is at least 89-~ 1 K m i n (5, el). A similar result holds for 89 2j+2 for k = 7, 8 . . . . , 12. 2j--2A-t

2j--2+t

2j--2+t\

Proof. T h e l e m m a h o l d s since t h e six 6 - v e c t o r s [ e 1 , 1 2 j ~ _ 1 , e l , 1 2 j + 2 , . . . , e1,12j+6) , t = 1, 2, . . . , 6, are l i n e a r l y i n d e p e n d e n t a n d since (j § 2, k), k > 1, are p a i r s w h e r e ej+z.k 2j'~l = 1. T h e last p r o p e r t y gives t h a t lie2j+1 ~- hll >> IleeJ+ll I + 0, 1 9 IIhL[ if h is a linear c o m b i n a t i o n o f v e c t o r s e, i <_2j 2 or i > 2j + 5.

and

We can assume that e 2j~2.

s, s 1 a n d

K

in L e m m a

1 a n d 2 are t h e s a m e for e2i+1

A BA~ACI:[ SPACE

~VITI-I B A S I S

CONSTANT

~

1

107

L]~MMA 3. L e t V 2 j - 1 a n d v 2j be as i n L e m m a 1. I f C is a subspace o f B , v ~j-~ E C a n d v ~j C C, a n d infgec llg - - e2J+llI = ~, then there is no p r o j e c t i o n f r o m the space D generated by C a n d e2j+~ onto C o f n o r m ~ 1 q - K . m i n (0, el) where K a n d e 1 are defined as i n L e m m a 2. A s i m i l a r result holds f o r e 2j+:. P r o o f . T h e l e m m a h o l d s since b y a n y p r o j e c t i o n f r o m D o n t o C t h e v e c t o r 89 2j+l is m a p p e d o n t o a v e c t o r w h o s e d i s t a n c e t o s o m e o f t h e e l e m e n t s O, (v 2j 1 _ (1 q_ 0.01(k - - 1))v"J)/Hv 2i-1 - - (1 q- 0.01(k - - 1))v2.iH, k : 1, 2 , . . . , 6, is 89-~ 89 m i n (~, ex).

prove that B has the basis constant ~p ~ I. We choose p = el) w h e r e e is t h a t o f L e m m a 1 a n d K a n d s 1 are t h e s a m e as in L e m m a 2. L e t D be a t w o - d i m e n s i o n a l s u b s p a c e o f B g e n e r a t e d b y v 1 a n d v ~, v 1 a n d v 2 as in L e m m a 1. I f E i s a s u b s p a c e o f B, D G E , such that there exists a p r o j e c t i o n f r o m B o n t o E o f n o r m ~ p , then by Lemma 3 E must c o n t a i n v e c t o r s w h o s e d i s t a n c e s t o ea a n d e4 are ~ e. T h u s b y L e m m a 1 a n d L e m m a 3 E m u s t c o n t a i n v e c t o r s w h o s e d i s t a n c e s t o e5 a n d e6 are ~ e. B y i n d u c t i o n w e see t h a t E m u s t c o n t a i n v e c t o r s w h o s e d i s t a n c e s t o e2j-~ a n d e 2j are ~ s for e v e r y j ~ 1. T h u s B h a s t h e basis c o n s t a n t ~ / 9 . We

now

1 q- 89

Remark. It has been shown by J. Lindenstrauss that the construction above can be m o d i f i e d so t h a t we g e t a u n i f o r m l y c o n v e x B a n a c h space i s o m o r p h i c t o H i l b e r t space with basis constant > I.

References 1. SIZ~GER, I., Bases in Banach spaces, Springer-Verlag, Berlin 1970.

Received March 27, 1972

Per Enflo Department of Mathematics University of California Berkeley, California U.S.A.