FARTHEST POINTS IN REFLEXIVE LOCALLY UNIFORMLY ROTUND BANACH SPACES BY
EDGAR ASPLUND ABSTRACT
If S is a bounded and closed subset of a Banach space B, which is both reflexive and locally uniformly rotund, then, except on a set of first Baire category, the points in B have farthest points in S. In the note [3] by Edelstein it is shown that, if B is a uniformly rotund Banach space and S a bounded and strongly closed subset of B, then the set
a(S) = { c ~ B : 3 s ~ S such that l l c - s t l > I I c - x l l V x ~ S } i.e., the set of all points in B that have farthest points in S, is dense in B. Here we will show, by a different method, that the restriction on B can be slightly relaxed and at the same time the condition on a(S) strengthened. We will require B to be reflexive and locally uniformly rotund (LUR), a condition first introduced by Lovaglia [4]: (LUR) Given e > 0 and x in B, with Hx I[ = 1, there exists a 6(e,x) such that
' -II (x + z)/2 II >for all ~ in B such that I1= II =< 1 and II x-=
'~
II-->="
Then a(S) will be shown to contain the intersection of a denumerable family of open and dense subsets of B. By the Baire category theorem, the intersection is itself dense in B. Such a set will be called a fat subset of B. We begin with a lemma on convex functions in arbitrary Banach spaces. LEMg_A 1.
I f f is a convex function which satisfies a Lipschitz condition
f ( x ) - f ( y ) <=cl]x - Yl] for all x and y in an arbitrary Banach space B, then there exists a fat subset E of B such that for each y in E and e > 0 there exist z in B and real numbers a, k satisfying Received July 19, 1966 213
214
EDGAR ASPLUND
[December
f<x) _<_a + k 11x - z I1' for all x in B , f ( y ) > a + k II Y - z I[2 - ~2/~.
Proof.
Let E n be the set of points y in B such that f ( y ) > a + k IIy - z 112 - X/nk
for some triple (z, a, k) in B x R 2 satisfying
f ( x ) < a + k 11x - z 112 for all x in B. Obviously, E, is open. If we can show that each En is dense in B, then o0
E = [") En n=l
will be fat in B and it will have the required property. To see that all E~ are dense take an arbitrary z in B and a k > 0 and put a = sup{f(x)-
kllx-
zll2:x~B}
In view of the Lipschitz condition on f i t is clear that, if k is large, the only points in B that influence the supremum above are those near to z. To be specific, we have a = sup{f(x) - k l l x - z l l 2 : x ~ n and tlx - zl[ < Ck -~ }. Choose first k sufficiently large and then y in B such that I1Y - z II -< Ck-1 and such that f ( y ) - k II y - z II2 + link > a. Thus E~ is dense in B and Lemma 1 is proved. REMARK. The above lemma is a variant of the method used in [1]. Suppose now that S is an arbitrary bounded subset of B. We then define a function r on B with positive values by the formula
r(x) = sup ~11x -s II:s ~ s) Also, we will use the notation
B(~, a) --- {y ~ B : II y - xll--< a ) for the " b a l l s " of B.
LEMMA 2. The function r is convex and satisfies the Lipschitz condition of Lemma 1 with C = 1. Moreover, for all y in B and b > O, sup {r(x): x e B(y, b)} = r(y) + b. Proof. r is the upper envelope of a family of convex functions, so it is convex, and it satisfies the Lipschitz condition because each of the members of the
1966]
FARTHEST POINTS IN ROTUND BANACH SPACES
215
family does so, by the triangle inequality. The last statement of the lemma follows from the sup inversion formula: sup{sup{l[x-sll:s~S~ X
IIx- ell =< b}
S
--
sup{sup{llx- sl[: Ilx - YII-<-b}: s~S}
--
sup{[~y-s[[ + b:s~S}
S
X
= r(y)+
b.
S
We will now go on to investigate the differential properties of a convex function f satisfying the conditions of Lemma 1, at the points of E . A subgradient o f f at the point y in B is an element p of B* such that
f(x) > f ( y ) + ( p , x - y )
for all x in B.
The set of all subgradients o f f at y is called the subdifferential o f f at y and denoted by a f ( y ) . By the Hahn-Banach theorem, Of(y) is a non-empty set for each y in B, namely, the continuity o f f implies that the set {(x,z):f(x)< z} is open in B x R and thus by the "geometric" version of the theorem (cf. Bourbaki [2], p. 69) separated by a hyperplane from the point (y,f(y)). LI~MMA 3. Let f satisfy the conditions of Lemma 1. Then if y is in E, the elements of Of(y) all have the same norm in B*. COROLLARY. Take f = r and y in E. Then the elements of t~r(y)all have norm
one. We may suppose that af(y) contains some elements p ~ 0, and we choose._< IIp Combining the inequalities, we get Proof.
(1)
[1/4.
(p,x-y)<e2/k+k([lx-z[12-llY-zll2)forallxinn.
Take the sup of each side of (1). For a given value of ] i x - y][, we get the most out of the right hand side of (1) by putting x - y = 2(y - z) for some 2 _>-0, then
'~ II p II II Y Now put 2 = ~/k
(2)
-
Ity - z II,
z
II --- e/k + k(2~ + z~)II y - ~ II ~.
the result is
II p II =< 2k II y - z II + 25.
If, instead, we take the inf of both sides of (1) and want to get the smallest value to the right of (1) for a fixed ]]x we put y - x = 2(y - z) and 0 -< 2 -< 1, because
yl[,
II x - ~ II >= II y - ~ II - II y - ~ II with equality in the above case. Thus
216
EDGAR ASPLUND
- ~ Irp II Ify - z l[ s 82/k + k ( - 2 ~ + ~2)11 y - z II~ Again put 2 = 8/k y - z , this is possible since by (2) and the condition
5 ~ IIp 11/4 we h a v e . ~ k IIy - z [I We get an inequality which combined with (2) gives (3)
[ I]P][-2k][y-z[I
[ < 28.
Since e > 0 can be chosen arbitrarily small, this proves Lemma 3. To see that the Corollary follows from Lemma 3, we use the estimate of Lemma 2 instead of the subgradient relation in (1) and proceed as before, the result is (2')
1 _-<2k HY - z I[ + 28
Combined with (3), this shows that l] P 11->- 1 whereas C = 1 in the Lipschitz condition for r proves the opposite inequality. We can now prove the statement announced in the beginning. THEOREM. I f the reflexive Banach space B is (LUR) and S is a bounded and closed subset of B , then a(S) is f a t in B. Proof. We will in fact prove that the set a(S) contains the earlier constructed set E for the function r on B. Suppose, then, that y is in E and take p in ar(y). The functional p assumes its minimum (since B is reflexive) on B(y,r(y)) at a point x. We will show that x is in S and hence a farthest point to y in S. With no loss of generality we assume that y = 0 and r ( y ) = 1, i.e., 1]x ]l = 1. Consider now the function r on the segment from 0 to - x . By the corollary to Lemma 3 the increase of r on this segment is exactly one. Hence the ball B ( - x , 2) is the smallest ball with center - x that contains S, so we may find points z in S very near to the boundary of this ball, i.e.
1 - I[(x + z)/2 II can be made arbitrary small. Say that it is made smaller than the quantity 6(e, x) in the condition (LUR); it then follows that l[ x - z [! < 5 because IIz 11<__1. But 5 can be chosen arbitrarily small, hence x is the strong limit of points z in S, so by hypothesis x is in S, as claimed. REFERENCES 1. E. Asplund, Frdchet differentiability of convex functions, To appear. 2. N. Bourbaki, Espaces vectorMs topologiques, Chap. I-II, Paris 1953. 3. M. Edelstein, Farthestpoints of sets in uniformly convex Banach spaces. Israel J. of Math. Vol. 4, (1966), 171-176. 4. A. R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78, (1955), 225-238. UNIVERSITY OF WASHINGTON,
SEATrLE,WASHINGTON
ON MULTIPLICITY THEORY FOR BOOLEAN ALGEBRAS OF PROJECTIONS BY L. T Z A F R I R I * ABSTRACT
The aim of this paper is to present a study of the connections between the commutants of a Boolean algebra of projections of finite multiplicity and the uniformly closed algebra generated by these projections. Dieudonn6 [4] has constructed an example of a Banach space X and a Boolean algebra (B.A.) of projections ~3 of uniform multiplicity 2 such that for no choice of xl and xz in X a n d 0 ~ Eel3 is2EX the direct sum of the cyclic subspaces spanned by Exl and Ex2. In this note, we shall prove that the first commutant of a B.A. of projections of finite multiplicity ~3, having Dieudonn6's above mentioned property (formal definition in Section 2), consists of those spectral operators whose scalar parts belong to the algebra 9~(~3) generated by ~3 in the uniform operator topology. However, we do not know if the nilpotent parts really exist. Later, using the previous result, we shall show that if there are no nilpotent operators commuting with a B.A. of projections of finite multiplicity ~3, then its commutant is commutative, i.e, coincides with the second commutant. Using another example of Dieudonn6 [3] we can conclude that it must not coincide with 9~(~3): 1. Preliminaries. For convenience we give here some definitions from Bade's papers [1] and [2].A B.A. of projections ~3 will be called complete if for every family {E,} _ ~ the projections V E, and A E , exist in ~3 and ( V E~)X = elm {E~X} ; (A E~)X = (") E~X A B.A. of projections will be called countably decomposable if every set of disjoint projections of ~ is at most countable. The cyclic subspace spanned by a vector x is defined by 9~R(x) = clm { Ex [E e ~3} Received April 22, 1966. * This paper is a part of the author's Ph.D. thesis to be submitted to the Hebrew University. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement. 217
218
L. TZAFRIRI
[December
If ~3 is a complete countably decomposable B.A. of projections in a Banach space X there exists a unique multiplicity function m defined on ~ such that m(E) is the least cardinal power of a set of cyclic subspaces spanning the range of E e ~ . The concept of spectral operator as used here is that which was developed by Dunford in [5]. Throughout the paper X denotes a fixed Banach space, ~ is a complete countably decomposable B.A. of projections of finite uniform multiplicity n i.e., every E ~ ~ has multiplicity n, and 9A(~) is the algebra generated by ~ in the uniform operator topology. Following [6], !Bc will be the commutant of ~ , i.e, the algebra of all operators commting with every E e ~ , and ( ~ c)c the second commutant of ~ , i.e., the algebra of all operators which commute with every operator commuting with ~3. Since ~ can be regarded as the range of a spectral measure E( • ) defined on the Borel sets of a compact Hausdorff space f~, to every Borel measurable function f we may consider the operator (in general unbounded) S(f)=faf(~)E(d~) with the domain P
D<s<:))-- I X,l J. where em=
exists
{~l lf(og){=<m}.
2. Operators commuting witli ~ . [4] we give the next definition
In connection with Dieudonn6's example
DErINmON 1. We shall say that !B is of type (D) if for no choice z~e X; 1 < i < n and 0 ~ e E ~ EX =
Ez i
Ez i
i
Li=p+I
;
l =
J
LEMMA 2. Every projection commuting with ~ belongs to it if and only if is of type (D). Proof. Assume 0 # P ¢ ~c is a projection and denote Fo =
V{EIF, ,EP=O }
F= I-Fo Since ~ is complete F ~ ~ and obviously PF = P. First, we shall show that Px = 0 implies Fx = 0. Indeed, if Pxo = 0 for some 0 ~ Xo ~ F X and
1966]
MULTIPLICITY THEORY FOR BOOLEAN ALGEBRAS OF PROJECTIONS
219
n
V ~l~(Fy,)
FX =
i=1
then )2 S(f~zem)Fy~
x o = lira m~oO
i=l
where em= {co [ co e l'~; [f~(og)[ < m; i = 1, 2, ..., n}; m = 1, 2 , . . . . For m suificiently large
0 ~ E(e,~)xo = ~ S(f~)E(em)Fy, l=1
and we may suppose that S(fn)E(em)Fy n ¢ 0. If e c em is any Borel set of positive measure on which fn satisfies the inequality (l/m) < If~(og)I < m then n--1
0 ~ E(e)Fy~ = S(f,-1 )E(e)txo - E S(f~f: l)E(e)Fy, i=1
and, therefore
E(e)FX = 9J~(E(e)Fxo) V 93~(E(e)Fy l) V "'" V ~R(E(e)F yn_ 1) In conclusion, there exist systems {0 ~ G e ~ ; G < F;xo, xl,...,x~_l} such that n--1
G x = V ~(Gx,). i=0
For each such system we can assume that xl,'",xn-1 were arranged such that
PGxo . . . . .
PGxk = 0;
k_>_0
and PGxi ~ 0 for k < i < n - 1. Now, from all those systems, let us choose one for which k is maximal. If k = n - 1, then PG = 0 which contradicts that
O¢ G> F;thusO< k
Vo~ V ~ ( a x 3 t=0
otherwise there exists 0 ~ G1 < G such that n-2
a l x = ~(G~vo) V V
~(alx3
t=O
and this fact contradicts the maximality of k. Thus V~=0~J~(Gx~) is the null space o f the restriction o f P to GX.
220
L. TZAFRIRI
[December
Now, let us consider all the systems {0 :/: H E ~ ; H < G;xo, xl,
"",Xk,,Xk+ l,
"",~n-1}
such that k
HX=
n-I
V 9~(Hx,)V V
i=k+l
i=O
~(H~7,)
and (I -- P)H.~k+.i = O; j = 1,...,l while ( I - P)H.~k+j#O for j = l + 1,..., n - 1 - k. From all these systems choose one for which l is maximal and assume that (I - P)wo = 0 for some Wo E HX. Then, by the same kind of argument one can easily see that k
k+l
Woe V 9Jt(Hxi) V i=O
V ~(H.~i)
i=k+l
and further k+l
Woe V ~(H~,) i=k+l
i.e., the null space o f / - P/nx coincides with v,k+l i=k+ 19J~(H~i). Hence H X can be decomposed as a direct sum as follows k
H X = V ffYC(Hxi)~ i=O
k+l
V ~(H.~i) i=k+l
where l > 1 since k < n - 1 and H has uniform multiplicity n. This contradiction shows that Xo = 0. Finally, suppose that P ¢ F. Then P x - F x :/: 0 for some x E X and P(Px - Fx) = 0; so we get a contradiction to the first part of the proof. Thus P = F ~ . The converse is obvious. Q.E.D.
Let ~ be of type (D). Every operator commuting with ~ is spectral and its scalar part belongs to 9A(f~). THEOREM 3.
Proof. By Foguel [7, Theorem 2.3 and Lemma 2.2], for every operator T E $ c there corresponds a sequence of Borel sets {am} increasing to f~ and such that
ZE(°~m) = ~ S(fix~,,,)ef,ra Jr Nrn ;
m = 1,2,...
i=1
T[I
where fi; i = 1,2,...,n are bounded measurable functions, [f/(og)[ < [I a.e.; Pl,m, Pz.m,'.',Pn.m disjoint projections commuting with ~ and Nm a nilpotent o f order n. By lemma 2 Pt, m E ~ ; i = 1,2,...,n; m = 1,2,..., thus
1966] MULTIPLICITY THEORY FOR BOOLEAN ALGEBRAS OF PROJECTIONS
TE(~,,) = S(g~) + Nm;
221
m = 1, 2,...
where gin; m = 1,2, ... are bounded measurable functions and Then, for k < m we shall get
Ig-<09>1--< IIz I1a.e.
TE(gk) = S(gmz~) + N,,E(~) hence { S(gm~Gk) = S(gk)
NmE(eR) = NR and further g,.(09) = gk(09) for almost every 09 ~ ek" Denote g(09) = gk(09);
o9 e ek ; k = 1, 2,.-.
Then g is a bounded measurable function and lim gin(09) = g(09) m'-* QO
for a.e. 09 e f~. By [6 Theorem, IV-10--10] lim S(gm)X = S(g)x ;
xeX
n l . - ~ O0
Now, if N = T - S(g) then
N x = lim Nmx;
xeX
rtl--~ O0
and, consequently N will be a nilpotent belonging to ~3c. In conclusion T = S(g) + N where S(g) e 9~(~) and N" = 0. Q.E.D. COROLLARY4. Let ~ be of type (D) such that ~ contains no nilpotent operator. Then ~Bc=(~3~)~= 9~(~3). It can be shown that there are no nilpotent operators commuting with the B.A. of projections constructed by Dieudonn6 [4]; thus both its commutants coincide with the algebra generated by the B.A. in the uniform operator topology. LEMMA 5. For any B.A. of projections of finite uniform multiplicity fB there exists 0 v~ Eo e ~ such that
EoX = X I~)"" ~ ) X k where Xj; j = 1 , 2 , . . . , k are subspaces invariant under fB; ~ restricted to X j has finite uniform multiplicity n j, (~k= 1 nj = n) and is of type (D). Proof. If ~3 is of type (D) the assertion is trivial. If it is not type (D), then we can find 0 # F e ~3 and z i e X ; i = 1,2, ..., n such that
f=l
i=p+l
222
L. TZAFRIRI
Among all these systems {F[0 # F e ~ ; minimal. Denote
[December
z l , . . . , z , } choose one for which p is
p
n
9J~(Fz,)
X1 = V ~IY~(Fz.,); Y, = V i=l
i=p+l
Then X1 and Y1 are subspaces invariant under ~ ; ~ restricted to X~ has finite uniform multiplicity p and ~ restricted to I11 has finite uniform multiplicity n - p. Furthermore, in view of the minimality of p, ~ restricted to X1 is of type (D). Repeating this process for Y1 and so on, wc shall finish the proof after a finite number of operations. Q.E.D. LE~IA 6. Assume there is no non-trivial nilpotent operator in fB c and let T ~ 0 be an operator commuting with lB. Then, there exists E o efB such that 0 # TEo e (~3c)~. Proof.
Denote
Fo =V{EIEe~;
TE--0}
F = I-Fo.
Then F E ~ and T F = T # O. Using Lemma 5 for F X we shall get a projection 0 # E o =< F; E o ~ such that EoX = X d ~ X 2 ( ~ ' " ( ~ X k
and ~3 restricted to X j; j = 1,2,..., k is of type (D). Let Pj be the projection on X i. Then k
T E o = ~, P i T j=l
and i f j # h PjTPh is evidently a nilpotent commuting with ~3; thus PjTPh = O. But P j T P j can be considered as an operator in Xi which commutes with ~3/x~. Hence by Corollary 4 PjTEo = P j T P j = S(fj)Pj;
j = 1, 2,..., k
where fj is a bounded measurable function. Consequently k
TEo = Y~ S ( f j)P j j=l
and, further, for another operator A which commutes with k
AEo = E S(gj)Ps. .I=1
1966] MULTIPLICITY THEORY FOR BOOLEAN ALGEBRAS OF PROJECTIONS
223
Thus k
(TEo)A = (TEo)(AEo) = • S(fj)S(gj)Pj = (AEo)(TEo) = A(TEo) j=t
i.e. TEoe (~3c)c. Finally, let us remark that TEo # 0 since 0 # E o < F. Q.E.D. THEOREM 7. Assume there is no non-trivial nilpotent operator in ~3c. Then =
Proof.
Let 0 # T ~ ~3*. Denote
and remark that by previous 1emma, ~ o is not void. If {E(/~)) is an increasing chain in ~3o then by [1 I g m m a 2.3]
A[T VE(a~)]x =
A T lim E(fi,)x = limATE(fi~)x 7
7
= limTE(fi~)Ax = [TVE(fi~)]Ax;
x~X, A~*
7
Thus VEr(~r) ~ ~3o and lemma of Zorn insures the existence of a maximal element E(flo) of ~3o. If TE(£~ - flo) # 0, then by Lemma 6 for the subspace E(fl - flo)X one can find E ( a o ) ~ 3 ; ao c f l - flo; 0 # TE(ao)~(~B~) ~. Consequently E(~o)VE(ao) belongs to ~3o which contradicts the maximality of E(flo). Therefore, T = TE(~o) ~ (~3~)~. Q.E.D. COROLLARY 8. L e t ~ b e a complete countably decomposable B.A. of projections containing no projections of infinite uniform multiplicity and such that there are no non-trivial nilpotent operators in ~ . Then ~ ~= (~)~. Proof. By [2 Theorem, 3.4] I = V oo__t E~ = ~ = 1E~ where E~ ~ ¢ are disjoint projections such that if E,, ~ O, it has uniform multiplicity. Hence, our statement follows from theorem 7. Q.E.D. 3. Remarks. a. Dieudonn~ [3] has constructed another example of a B.A. of projections ~ of finite uniform multiplicity (n = 2) for which there are no non-trivial nilpotent operators in its first commutant and 9~(~) is a proper subalgebra of ~ c = ( 3c)*. It shows that in Theorem 7 ~B*=(~*) * must not coincide with 9/(~3). b. In the decomposition of Tgiven byTheorem 3 a nilpotent part was obtained. We do not know if it really exists. The single example of a B.A. of type (D)which has been constructed until now is that of Dieudonn6 [4] and one can easily show that there is no nilpotent commuting with Dieudonn6's B.A. of projections.
224
L. TZAFRIRI
I f this is the general case and the c o m m u t a n t o f a B.A. o f projections o f type (D) always coincides with the algebra 9~(~), then we are able to prove the converse o f Corollary 8, i.e. commutative o f the c o m m u t a n t implies the absence o f nontrivial nilpotent operators in the first c o m m u t a n t . REFERENCES 1° W. G. Bade, On Boolean algebras of projections and algebras of operators, Trans. Amer.
Math. Soc. 80 (1955), 345-359. , A multiplicity theory for Boolean algebras of projections on Banach spaces, Trans. 2. - Amer. Math. Soc. 92 (1959), 508-530. 3. J. Dieudonn6, Sur la bicommutante d' une alg~bre d' op~rateurs, Portugal Math., 14 (1955), 35-38. , Champs de vecteurs non localement triviaux, Arch. Math., 7 (1956), 6--10. 4, ---5. N. Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321-354. 6. N. Dunford and J. Schwartz, Linear operators, 1, New York Interscience Publishers, 1958. 7. S. R. Foguel, Boolean algebras or projections of finite multiplicity, Pacific J. Math., 9 (1959), 681-693.
THI~ H E B R E W UNIVERSITY OF .JERUSALEM
SETS WITH HOLES O N THE REAL LINE: A C O N S T R U C T I O N A N D A N APPLICATION BY J U A N JORGE SCH~.FFER ABSTRACT
An increasing sequence of closed subsets of the unit interval is constructed, such that no set in the sequence is essentially covered by a finite union of translates of the set preceding it. The result is applied to construct a pathological right-translation-invariant ideal of the Banach lattice L ¢° on R+. 1. Introduction. In [3] a conjecture on the existence of very " t h i n " measurable sets on the real line was settled in the affirmative by the trick of constructing certain sets with evenly scattered holes. Since both the size and the distribution of the holes could be closely adapted to the purpose in hand, it was tempting to put such sets to further uses. This note illustrates another such use. The question dealt with in [3] originated in a problem of the theory of function spaces with translations on the positive half-line, as developed in [2] (cf. [1], Chapter 2); so does the question discussed here. In Section 3 we describe this application without technical dependence on those references. 2. The construction. R denotes the real line and/~ is Lebesgue measure on R. If F, F ' c R are #-measurable, we say that F is essentially contained in F ' if ~,(F / F') = O. THEOREM 1. There exists a sequence (E,n) of closed subsets of [0, 1] such that # ( E l ) > 0 , E,ncEm+1, and E,n+t is not essentially contained in any finite union of translates of E m, m = 1,2, .... Proof. 1. We define a positive-integral-valued function p(m;n) on the pairs of positive integers, by recursion in the first variable, thus: (1)
p(l;n) = n+l,
n=1,2,.-.;
if p(m; n) is known for a given m and all n we define the following auxiliary function by recursion in j : (2)
q(m,O;n)=n; q ( m , j + l ; n ) = q ( m , j ; n ) + l + p ( m ; q ( m , j ; n ) + l ) , j = 0,...,n-I,
and finally set Received September 6, 1966. 225
n = 1,2,...
226
JUAN JORGE S C ~ F E R
[December
(3)
p(m + 1; n) = q(m, n; n) - n + 2.
It is immediately verified by induction that p(m;n) is positive-valued and that (4) 2.
p(m; n) is strictly increasing in both variables.
For every positive integer m, we define the closed set oo
A m = R \ LJ /1=2
o0
LJ (2-"r + (0, 2 -"-p(-;") )). r=--oo
The open intervals 2-% + (0,2-"-v(m;/1)), r = 0, _+ 1, + 2, ..., are the holes of order n of Am; the images of these holes under the translation t ~ t' + t are the holes of order n of t ' + A m. On account of (4), A m c A , , + I for all m. We set Em= [0, 1] n Am, so that (Era) is a sequence of closed subsets of [0,1] with E m c E m + l . Using (1), we find oo
p(E1)>I-
~
2n--1
~E 2 -"-v(1;")= 1 -
n=l
r=O
oO
~ 2 -"-2 = ½ > 0 . n=l
To conclude the proof, we show that, for any given m, Em+~ is not essentially contained in any finite union of translates of Am, let alone of Era. 3. More specifically, we claim that for any given positive integers m, k and any tl, ..., t k ~ R we have k
(5)
# ( [ 1 - 2-k, 1] r~Em+,) > p ( [ 1 - - 2-~,1] N iU=l(t,+Am)),
which implies the desired conclusion. Indeed, since p(m + 1 ; n) > p(1; 1) = 2 for all n, [1 - 2 -k, 1] does not meet any holes of Am+ 1 of any order strictly smaller than k; and it contains, for each n > k, 2 n - k holes of order n and total m e a s u r e 2n-k2 -n-v(m+l; n) = 2 - k - v ( m + l ; n) < 2 -"-v<m+l;k) (on account of (4)). Therefore (6)
p([1-2-k,
1] x E m + l ) = # ( [ 1 - 2 - k ,
1]nAm+l)
oo
> 2-~ _ ~ 2-"-v("+l;k) = 2-k(1 _ 21-v(m+1;~)). n=k
On the other hand, [1 - 2 -k, 1], being of length 2-*, contains at least one interval of the form 01 + 2 - k - l r , tl + 2 -~- ~(r + 1)) (r an integer), and therefore contains one complete hole of t, + A., of order k + 1 and length 2 -C*+*)-v°":*+*) = 2-q("'*;k); this hole again contains a complete hole of t2 + A , . of order q(m, 1;k) + 1 and length 2 -q(m'2;*) (we here use (2)); continuing in this fashion, we find that [ 1 - 2 -k, 1] contains an open interval of length 2 -~(''k;*) that is
1966]
SETS WITH HOLES O N THE REAL LINE
227
a hole of tk + A= contained in a hole of tk_ 1 + Am'." contained in a hole of tx + A=, and thus disjoint from [.JR 1 (q + At,). Therefore, using (3), k
/~([1 - 2 - k ] n ~=,= (fi +
Am)) =< 2 -k -- 2 -q(''k:k) = 2-k(t
- 22-p(m+l;k)).
This, together with (6), implies (5), thus establishing our claim. 3. The application. R+ denotes the positive real half-line. Measure, measurable, null, all refer to Lebesgue measure. L~ is the usual Banach lattice of (equivalence classes modulo null sets of) essentially bounded measurable real valued functions on R+, with the essential-supremum norm I]" I]" A constant in L =, is denoted by its constant value, the characteristic function of the measurable set E c R + by X~. For f e L °°, s e R, we define the translate T~fe L °° by Tj(t)
Yf ( t [ 0
s)
t > max{0, s}
0 <=t < s
(with the usual measure-theoretical grain of salt). F = L°° is an ideal in L°° if F is a linear manifold satisfying
(F): T F,
Igl--Ifl
An ideal F c L ~° is right-translation-invariant satisfies (RT) [(T)]: f e F ,
seR+[seR]
implies g ~ F . [translation-invariant]
if it
implies T f f e F .
It is an easy exercise to show that the L%closure of a right-translation-invariant ideal is another (the same is true, for that matter, for translation-invariant ideals). If F is a right-translation-invariant ideal, the smallest translation-invariant ideal containing F is, clearly, T - F = { T - s f : f ~ F , s ~ R+}. THEOREM 2. There exists a closed right-translation-invariant ideal F in L °~ such that T - F is not closed in L °°. Proof. Let (E,,) be as in Theorem 1, and set B = [.J~= l(m + E,,), a closed set in R+. Define G as the set of those f e L~ that vanish a.e. outside some finite union of right-translates of B; thus f e L °~ with [If]l ==_a is in G if and only if If[ <_-zc, where C = [,,Jk=1(s , + B) for some positive integer k and numbers s~eR+, i = 1,...,k; in particular, if E c R+ is measurable, Xre G if and only if E is essentially contained in such a set C. We claim that this is specifically not the case for E = p + Ep+a for any positive integer p ; otherwise, indeed, p + Ep+l would be essentially contained in C = [,.J~=, [,.J ,k_-l(m + s, + E,,); since (p + Ep+l) n ( m + s, + E,,) is null for
228
JUAN JORGE SCH~-'FER
[December
m > p + 1, and E m c Ep for m < p, Ep+l w o u l d - i n R of c o u r s e - b e essentially contained in l,.J,n= P l U~= k I (-- P + m + st + Ep) , contradicting the conclusion of Theorem 1. G is obviously a right-translation-invariant ideal, so that its L°°-closure, which we choose as our F , is another.
II--
Now Zm+~. ~ G, so that XE, = T-,Xm+ ~,, e T - F , m = 1, 2,.-., with IIX~, 1. Therefore ( Z ~ = 12-'~Zg_) is an L~-Cauchy sequence in T - F , and its L%limit is fo = ~ = 12-mXE,, < Xt0,11" We claim that fo is not in T - F , thus proving that T - F is not L%closed. For assume that fo ~ T - F , i.e., fo = T - s f for some f e F , s e R + ; choose a positive integer p > s, and set E = p + Ep+I; then 0 < Tffo = T~T-sf < and therefore
Ifl,
0<2-P-~z~=
Tp(2-P-IZe~+,) < Tvfo=Tp_sTffo <
Zp_slfl e,
whence z E s F . There exists, therefore, g e G such that ]XE-- gl < IIz -g ---<½, whence Zg - zgg < ZE ] ZE - g[ < ½Ze, which implies [g I > zeg > ½Zt > 0. Thus XE~ G; and this was shown above to be impossible. REMARK. Theorem 2 provides a negative answer to the question in [2], p. 237 (following Theorem 4.6) and justifies the statement in [1], p. 61 (following 23.H); it is sufficient to provide F and T - F with the norm of L ~°. Our proof of Theorem 2 used the "local structure" of R+ very strongly, via Theorem 1. This prompted the author to raise the question concerning the analogous problem for the sequence space l °~ (on the set Z+ of non-negative integers). The referee has indicated the proof of the following analogue of Theorem 2; it may be adapted in an obvious way to yield an alternative proof of Theorem 2 itself. The meaning of the terms will be clear without fresh definitions. TUEORnM. 3. There exists a closed right-translation-invariant ideal F in 1~ such that T ~ F - i s not closed in l °~. Proof. Let a, be the product of the first n primes and set E~ = {a~: h = 1, 2,...} ~ Z + , n = 1,2,-... Let G be the set of those f e l °° that vanish outside some finite union of right-translates of 1 + E l , 2 + E 2 , . . . , and let F be the /%closure of G. F if a dosed right-translation-invariant ideal. As in the proof of Theorem 2, ( ~ , = ~ 2 - r ' x E , ) is an l°%Cauchy sequence in T - F ; if its/%limit fo ~m°°=12-rnxEm were in T - F , it would follow as in that proof that Zp+E~+,eG for some p ~ Z + ; but this means that p+Ep+~ I,.Jk= 1 (ri + i + Ei) for some positive integer k and rt ~ Z + , i = 1,..., k. This, however, is impossible, for (p + Ep+l)r~(r~ + i + E~) is a finite set for each i: J indeed, if p + a p +h 1 = r ~ + i + a i h' , the integer is divisible by aq, where q = min {i, p + 1}, j = rain{h, h'}; thus either h or h' is bounded, and the =
Ir~+i-pl
1966]
SETS WITH HOLES ON THE REAL LINE
229
intersection is consequently finite, unless p = ri + i; but in this last case, (p d- Ep+l) l'3 ( r i + i + Ei) -~ p + (Ep+ 1 (3 E~)is empty, since i < r~ + i = p < p + 1. The author wishes to thank the referee for suggesting the inclusion of the preceding result. Part of the research in this paper was carried out while the author was visiting professor at Carnegie Institute of Technology, Pittsburgh, U.S.A. REFERENCES 1. J. L. Massera and J. J. Sch/iffer,Linear differential equations and function spaces, Academic Press, New York, 1966. 2. J. J. Sch~iffer, Function spaces with translations, Math. Ann. 137 (1959), 209-262. 3. - - , P r o o f o f a conjecture about measurable sets on the real line, Proc. Amer. Math. Soc. 13 (1962), 134-135. ! UNIVERSIDAD DE LA REPUBLICA, MONTEVIDEO, URUGUAY
ON THE
SHARPNESS
OF MEIER'S
OF FATOU'S
ANALOGUE
THEOREM
BY
FREDERICK BAGEMIHLQ) ABSTRACT
Meier's topological analogue of Fatou's theorem is shown to be sharp by exhibiting a bounded holomorphic function in the unit disk for which no point of a prescribed set of first category on the unit circle is a Meier point. Let F be the unit circle and D be the open unit disk in the complex plane. We say that almost every point of F has a certain property, provided that the exceptional set is a subset of F of Lebesgue measure zero. Similarly, we say that nearly every point of F possesses a certain property, provided that the exceptional set is a subset of F of first Baire category. The celebrated theorem of Fatou (see [5, p. 5]) asserts that if f (z) is a bounded holomorphic function in D, then f has an angular limit at almost every point of F. Lusin and Priwaloff [-3, pp. 156-159] (see also [2] and [6]) have shown that Fatou's theorem is sharp by proving that if E is a subset of F of measure zero, then there exists a bounded holomorphic function in D that has no asymptotic value at any point of E. Meier [4, p. 330, Theorem 6] has recently obtained a topological analogue of Fatou's theorem that may be formulated as follows. If f(z) is a function defined in D, and if ( ~ F, then the cluster set of f at ~ is denoted by C(f, ~) (the rudiments of cluster-set theory are to be found in [-5]). The chordal principal cluster set o f f at ~ is defined to be
]--L (f, ~) = (~ Cx(f, ~), X
where X ranges over all chords (of the unit circle) at ~ and Cx(f, ~) stands for the cluster set of f at ~ along X. We call a point ~ ~ F a Meier point of f , provided that
(1)
0 = c(f, 0 = n,,
where " c " symbolizes proper set inclusion and f~ represents the Riemann sphere. Meier's theorem asserts that if f(z) is a bounded holomorphic function in D, then nearly every point of F is a Meier point of f . Received July 22, 1966. (x) Supportedby the U. S. Army Research Office,Durham. 230
1966]
ON MEIER'S ANALOGUE OF FATOU'S THEOREM
231
Before considering the question of the sharpness of this theorem, let us remark that Meier's theorem is not a trivial consequence of Fatou's. For if a point ~ e F at which f has an angular limit is called a Fatou point o f f , then there exists in D a bounded holomorphic function of which nearly every point of F is a Meier point but not a Fatou point. An example of such a function is a Blaschke product b(z) with the property [1, p. 1070] that, for nearly every point ~ e F, c~(f, 0 = c ( f , 0 = r u O ~ f l ,
where Cp(f, 0 is the radial cluster set of f at ~. Our aim is to prove the following THEOREM. Let E be a subset of F of first category. Then there exists a bounded univalent holomorphic function f(z) in D such that no point of E is a Meier point o f f . Proof.
Since E is of first category, we may write E = Et
LIE 2 U...
UE. U-..,
where each En is a nowhere dense subset of F. Denote the closure of E, b y / ~ , and define
~' = E, uE~ u - . . u E . u . . . . Since each En is also nowhere dense, the set E' is an F~ of first category. According to Lohwater and Piranian [2, p. 7, Theorem 1'], there exists a bounded univalent holomorphic function f(z) in D with the property that at every point ( c F, f(z) has a radial limit, call it f ( O , and f ( O , regarded as a function of ~ along F, is discontinuous at every point of E' (and is continuous at every point of F - E'). Now it is seen directly that if f (z) (z ~ D) has a global limit at a point ( e F, then ( is a point of continuity of the function f(() (( E F). Consequently f(z) does not have a global limit at any point (~ E'. (On the other hand, according to Weniaminoff [7, p. 92, Lemma 2], f(z) has the global limit f(() at every point ~ F - E'). This means that for every ( e E' we have
cAf,~)
= { f ( O ) = c ( f , ~),
and hence
l-]~(f, () ~ c(f, O, so that (1) is not satisfied, and therefore ~ is not a Meier point off. Since E __ E', the proof of the theorem is complete.
232
FREDERICK BAGEMIHL REFERENCES
I. F. Bagemihl and W. Seidel, A general principle involving Baire category, with applications to function theory and other fields, Proc. Nat. Acad. Sci. U.S.A., 39 (1953)7 1068-1075. 2. A. J. Lohwater and G. Piranian, The boundary behavior of functions analytic in a disk, Ann. Acad. Sci. Fennicae A I, 239 (1957), 1-17. 3. N. Lusin and J. Priwaloff, Sur l'unicit6 et la multiplicit6 des fonctions analytiques, Ann. Sci. l~cole Norm. Sup. (3), 42 (1925), 143-191. 4. K. Meier, Ober die Randwerte der meromorphen Funktionen, Math. Ann., 142 (1961), 328-344. 5. K. Noshiro, Cluster Sets, Berlin, 1960. 6. W. Schneider, On the impossibility of sharpening the Fatou radial limit theorem (to appear). 7. V. Weniaminoff, Sur un probl~me de la repr6sentation conforme de M. Carath6odory, Recueil Math. Soc. Math. Moscou, 31 (1922), 91-93. UNIVERSITYOF WISCONSIN-MILWAUKEE MILWAUKEE,WISCONSIN
ON
CLIQUES
IN GRAPHS BY
P. ERDOS ABSTRACT
A clique is a maximal complete subgraph of a graph. Moon and Moser obtained bounds for the maximum possible number of cliques of different sizes in a graph of n vertices. These bounds are improved in this note. Let G(n) be a graph of n vertices. A non empty set S of vertices of G forms a complete graph if each vertex of S is joined to every other vertex of S. A complete subgraph of G is called a clique if it is maximal i.e., if it is not contained in any other complete subgraph of G. Denote by g(n) the m a x i m u m number of different sizes of cliques that can occur in a graph of n vertices. In a recent paper [1] M o o n and Moser obtained surprisingly sharp estimates for g(n). In fact they proved (throughout this paper log n will denote logarithm to the base 2) that for n >_-26 (1)
n - l-log n] - 2[-loglog n] - 4 < g(n) < n - [log n]
In the present note we shall improve the lower bound on g(n). Denote by lOgk n the k-times iterated logarithm and let H(n) be the smallest integer for which lOgrlt,)n < 2 . Let n 1 = I n - l o g n H(n)] and for i > 1 define n~ as the least integer satisfying (2)
2"' + n i - 1 > n~_ 1.
N o w we prove the following TI-IEOREM. g(n) > n -- log n -- H(n) - 0(1). H(n) increases much slower then the k-fold iterated logarithm thus our theorem is an improvement on (1). It seems likely that our theorem is very close to being best possible but I could not prove this. In fact I could not even prove that lim (g(n) - (n - log n)) = oo. .mOO
The p r o o f of our theorem will use the method of M o o n and Moser [1]. We construct our graph G(n) as follows: The vertices of our G(n) are x l , . . . , x , 1 ; Y l , " ' Y , 2 ; z x " ' z , , , where n l = [ n - l o g n - H ( n ) ] , n2 is defined by (2) and m = n - n 1 - n 2. Clearly m = H ( n ) + 0(1). Any two x ' s and any two y ' s are joined. Further for 1 < j < n 2 yj is joined to every x~ except to the x~ satisfying Received November 15, 1966. 233
234
P. ERDOS 2J-l+j--
[December
2 < i =<2i + j - -
1
and Y,2 is joined to every x~ except to those satisfying 2 "2-1 + n2 - 2 < i =< n x (n I =< 2 "~ + n 2 - - 1). N o w we use the vertices ZR, 1 <-- k <- m, z k is joined to yj for 1 ~ j <=nk +2 and to the xi for 1 <- i <_ nk+ 1. N o two z ' s are joined. This completes the definition o f o u r G(n). N o w we show that our G(n) contains a clique for every (3)
n,.+2 < t _<_n 1
and since by m = t t ( n ) + 0(1) and (2) n,,+2 is less than an absolute constant independent o f m, (3) implies our Theorem. Assume first n2 = t < nl. F o r t = n 2 the set o f all y ' s and for t = nl the set o f all x ' s gives the required cliques. F o r nl < t < n2 we construct our clique o f t vertices as follows: We distinguish two cases. If nl - t < 2 "2-1 we consider the unique binary expansion nl - t = 2 jl + ... + 21" , 0 =<Jl <: "'" < J , < n2
-
-
1.
I f 2 "2-1 _--
NOTES ON KLEE'S PAPER "POLYHEDRAL SECTIONS OF CONVEX BODIES" BY
JORAM LINDENSTRAUSS* ABSTRACT
In sections 2 and 3 two methods for proving the non existence of certain universal Banach spaces, are presented. In section 4 it is proved that every infinite-dimensional conjugate Banach space has a two-dimensional subspace whose unit cell is not a polygon. 1. Introduction. A Banach space X is called universal for a class of Banach spaces if every member of this class is isometric to a suitable subspace of X . S. Mazur has raised the question whether for a given integer j > 2 there is a finite-dimensional Banach space which is universal for the class of all j-dimensional spaces. This question was solved negatively for j = 2 (and hence for every j > 2) by C. Bessaga [1]. He showed that the set of all subspaces of a finite-dimensional space cannot include all two-dimensional spaces since it has a too low dimension. (Since we do not use this fact here we do not enter into the precise definition of the notions involved in it.) Klee [4] has extended considerably the argument of Bessaga and obtained for example numerical estimates for the dimension of Banach spaces universal for all j-dimensional spaces whose unit cell is a polyhedron with r vertices. Our first aim in this paper is to present a different method for proving the non-existence of certain kinds of universal finite-dimensional Banach spaces. Our method seems to be conceptually more elementary than that of Bessaga and Klee. We exhibit for every n a set of 2" polyhedral 3-dimensional Banach spaces such that no Banach space of dimension < cn/log n has all those spaces as subspaces. Our aim is mainly to present the method and we have not tried to get sharp estimates. From the results of Bessaga and Klee it follows by a routine compactness argument that for every j and n there are k(j, n)j-dimensional spaces such that no n-dimensional space is universal for them. It seems to us that from the point of view of estimates for k(j, n) our method will probably give better results than those obtainable by [4]. We have not checked Received October 7, 1966. * The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force. 235
236
JORAM LINDENSTRAUSS
[December
this and moreover we are sure that also the estimates of k(j,n) which can be obtained by our method are rather crude. The method used here is closely related to the one used in [7, pp. 98-99], for proving a result on the extension of operators. As far as Mazur's problem is concerned, this method has one obvious disadvantage in comparison with Bessaga's. It cannot be used in the case j = 2, it works only for j > 3. It is hoped that the method can be modified so as to solve some open problems concerning infinite-dimensional universal spaces (see section 5(a)). At the end of his paper [4] Klee mentions some infinite-dimensional questions. One of the questions (originally posed by S. Mazur) is whether there exists a separable reflexive Banach space which is universal for all separable reflexive spaces. This problem was solved negatively by W. Szlenk (private communication 1965). Szlenk showed in fact that there is no Banach space X with a separable conjugate such that every separable reflexive space is isomorphic to a subspace of X. We present here an extremely simple solution of Mazur's problem. Our method is not powerful enough to derive Szlenk's result but it applies in many situations in which Szlenk's method does not apply. Section 4 is devoted to the solution of another problem mentioned at the end of [4]. Klee called a Banach space X polyhedral if the unit cell of every finitedimensional subspace of X is a polyhedron. He showed that Co is polyhedral and asked whether there is an infinite-dimensional reflexive polyhedral space. By using a method due to Klee himself [6] we prove in section 4 that every infinite-dimensional space has a two-dimensional quotient space whose unit cell is not a polygon. Hence no infinite-dimensional conjugate space is polyhedral. We conclude the paper with a section devoted to open problems. All Banach spaces are taken over the reals. Let X be a Banach space. We denote by Sx(xo, r) the cell {x; IIX - X o II < r}. The unit cell Sx(O, 1)is denoted also by
Sx. Section 2.
We bring first three simple and well known lemmas.
LEMMA 2.1. Let X be a Banach space and let {Sx (x~,ri)} be a family (finite or in[inite) of mutually intersecting cells in X . Then there is a Banach space Y = X with d i m Y [ X = 1 and a point y e Ysuch that [ l Y - x, it <=r,for every i. This lemma is due to Nachbin [8]. Two simple proofs of it may be found in [7, p. 51]. The second of these proofs (due to Griinbaum) shows that if Sx has 2h extreme points and if there are k given cells than Y can be chosen so that Sr has at most 2h + 2k extreme points. LEMMA 2.2. Let X be an m-dimensional Banach space (m < ~ ) . Le~t t5 > 0 and let {xi},= k 1 be a set of points in Sx such that IIx, - x j tl >=2fi for i ~ j . Then k ~ (1 .-[- £~-1)m.
1966] ON KLEE'S PAPER "POLYHEDRAL SECTIONS OF CONVEX BODIES"
237
Proof. The interiors of the cells Sx(xi,6) are mutually disjoint and all are contained in Sx(O, 1 + 6). By considering the volumes of the cells the result follows. LEMMA 2.3. Let B(n), n > 3, be the two-dimensional Banach space whose unit celt is the regular 2n-gon. Then there are vectors {bi}~= 1 in B(n) such that llbill=2n2/n 2 for every i and I l b , + b j l l < = l [ b , H + l l b j [ I - 2 for i # j . Proof. Take n consecutive sides of the boundary of the unit cell of B(n). Let bi be the vector of norm 2n2/~ 2 in the direction of the middle of the ith side. Then for i ~ j ,
tl b,___b,I [ __<4n2n-2cos2rc/n < 4n2n -2 -
2.
We are now ready to prove Tm~OREM 2.1. Let n be an integer > 3. Then there exist 2" three-dimensional Banach spaces {Co} such that for every O, Sco is a polyhedron with 4n + 2 vertices and such that there exists no m-dimensional space which is universal for all the C o if 2" > 3m(2n 2 + 1) TM . Proof. Let B = B(n) and {b~}~'=1 be as in Lemma 2.3. For every choice of n signs 0 = (Oa,Oz,...,O,) , i.e. 0 i = _+ 1 for every i, there is by Lemma 2.1 a Banach space C o = B and a vector uo•Co such that Itu0t] < 1 and 11uo-O,b, b, 1 for e v e r y / . By the proof due to Griinbaum of Lemma 2.1 we can take as C o a space whose unit cell has at most 4n + 2 extreme points. We may even assume that Sco has exactly 4n + 2 extreme points (otherwise approximate the unit cell by a polyhedron with 4n + 2 vertices and use such an approximation as a new unit cell. It will be evident that by doing this we do not effect the argument below).
II-<--II II-
Choose in B two vectors y , z with IlYII=I z = 1 such that [12Y+PZ]I < m a x ( ] 2 ] , ] #l) for all2 and/~. We have that b i = 2,y + / h z with 2 i , 2n2/n2 for every i. Let X be a Banach space of dimension m which is universal for all the Co and consider all possible isometric embeddings of B in X . By Lemma 2.2, with 6 = 1/(2n2), there exist k < (2n 2 + 1) 2misometric operators Tj: B ~ X , j = 1,...,k, such that for every isometry T : B ~ X there is a j such that (2.1)
1]T y - T y 1] < n - 2 '
]1Tz - Tjz H < n-2
Take now two different n-tuples of signs 0' and 0". Let T ' and T" be isometrics from Co, and Co,, respectively into X and assume that there is a common j for which (2.1) holds for the restrictions of T' and T" to B. Since 0~ ~ 0[ for some i we may assume without loss of generality that 0'I = 1 and 0£'= - 1. Then
238
JORAM LINDENSTRAUSS
[December
[I T'u o, - Tjb 1 II -< II Z'uo, - T'bl ][ + I1T'bl - Tjbl II -<
<=
I[Uo,-blil+2"2n2~-2.n-2<=l[blll-1+4~-2
Similarly II z"u0 + r~b~ I! < II82 I[- 1/2 Since IIZjbl ][--II bl il we get that 1[T ' u o " - T"uo,, I[ > 1. Hence, by Lemma 2.2, there are at most 3" different 0 for which there is an isometry To:Co ~ X such that the restriction of To to B satisfies (2.1) for a given j (observe that [1Touo II --< 1 for every 0 and use the lemma with 6 = 1/2). Since the number of the indices j is k < (2n 2 + 1) 2m we get that 2" < 3"(2n 2 + 1) 2'n. Q.E.D. Section 3. We present now a simple method for proving the non-existence of certain infinite-dimensional universal spaces. If X and Y are Banach spaces and l < p < o o we denote by ( X @ Y ) p the space of all pairs (x,y) with
II~x, y)I[ -- (11x I[p + IIY 1[')I/p if p < oo and = max(] I x II, I! y and y e Y). The one dimensional space is denoted by R.
II) ir p = ~ (x ~ x
LEMMA 3.1. Let X be a Banach space and let 1 < p < oo. Assume that (X O)R)p is isometric to a subspace of X . Then X has a subspace isometric to Ip if p < ov and to co if p = o o . Proof. We consider the case p < oo only, the proof for p = oo is similar. By our assumption X contains a vector x 1 with xl ] = 1 and a subspace Y1 which is isometric to X such that x 1 + y p = 1 + y p for every y E I71. Continuing inductively we get for every n a vector x, in Y,_ a with x, = 1 and a subspace Y, of Y,-1 which is isometric to X and such that IIx, + y ~ = 1 + I y IIp for every y ~ Y,. It follows easily that for every real {2~)7=x we have " [~,I p" Hence the subspace of X spanned by {x ,},=1 oo has 1[ Z ,n= I 2ix, ][P = ~,=1 the desired properties. TrrEOR~M 3.1. The following sets of Banach spaces do not have a member which is universal for all the spaces in this set. (i) All reflexive spaces of a given density character .,¢g. (ii) All spaces whose conjugate is separable. (iii) All conjugate separable spaces. (iv) All spaces of a given density character ~ which do not contain an infinite-dimensional reflexive subspace. Proof. All these facts follow easily from Lemma 3.1. (i) follows since Co and 11 are not reflexive. (ii) is a consequence of the fact that l* is not separable. To derive (iii) we have to use a result of Bessaga and Petczyfiski [2] that no separable conjugate space contains a subspace isomorphic to Co. Finally, (iv) follows by using the lemma for 1 < p < oo. Section 4. In this section we prove that there is no conjugate infinite-dimensional polyhedral space. First some notations and conventions. Let P be a sym-
1966] ON KLEE'S PAPER "POLYHEDRAL SECTIONS OF CONVEX BODIES'"
239
metric convex body in the plane R 2 . The boundary of P will be denoted by 0P and for 0 < e < 1 the interior of the set (1 + e)P ,,, ( 1 - e)P will be denoted by nbd 8P. We shall consider in the sequel many norms in R 2 . However, for the sake of definiteness we single out one norm (an inner product norm, say) and this will be used whenever the symbol II[I is used for a vector in R z or an operator into R 2 . All other norms in the plane will appear only implicitly through their unit cell and the symbol I1" I1 will not be used for them. If Y c X and T is an operator on X its restriction to Y will be denoted by Tit. -
LEMMA 4.1. Let X be a finite-dimensional Banach space whose unit cell is a polyhedron. Let e > 0 and let T be a linear operator f r o m X onto R 2. Assume that there is a subspace Y of X of co-dimension 1 such that T S x = T S r . Then there is an operator T : X . . . , R 2 such that a~Sx~ e - nbdaTS x and such that T S x has more extreme points than T S x .
[Izl,- lYlt<= ,
Proof. Let {___ci}~= 1 be the extreme points of T S x and put Ai = T - lc i n S x, i = 1,2,...,n. Assume first that there is an index i = i o such that Aid consists of more than one point. Let f e X* be such that f takes on A~o positive and negative values. Let c ~ R 2 be such that {C~o+ 2c; 2 s R} n T S x = C~o and put Tox = T x + Of(x)c. It is easily verified that for small enough 0 the operator To has all the properties required from T. Assume now that A~ is a single point for every i. By our assumptions A~ s Y for every i. For fi > 0, let T~ be an operator from X into R 2 such that T~ is one to one on extS x and II Zll < ~ it is easily checked that for small enough 6 the points {___T~Ai)~= 1 are extreme points of T~Sx. Take one such 6 = rio < e/2. If TaoSx has more than 2n extreme points we can take T = T~o. If {_ T~oA~),"=~ are all the extreme points of T~oSx then by the argument of Klee in [6] there is a ~:X--~ R 2 such that T i t = Too It, TSx has more than 2n extreme points and 8TS x ~ el2 - nbdOTooS x. This operator 7~ has all the desired properties. LEMMA 4.2. Let P be a symmetric polygon in the plane. Then there is e > 0 such that every symmetric convex body C in the plane f o r which aC ~ e - nbdOP has at least as m a n y extreme points as P. Proof.
Obvious.
THEOREM 4.1. Let X be an infinite-dimensional Banach space. Than X has a two-dimensional quotient space whose unit cell is not a polygon. Proof. We assume that X is an infinite-dimensional Banach space all whose two-dimensional quotient spaces are polygonal and argue to a contradiction. By the results of [5] every finite-dimensional quotient space of X is polyhedral. Let T~ be a bounded linear operator from X onto R 2 and let T~Sx have 2n~ extreme points. Let T I : X ~ R "'+~ and U1 :R " ~ + ~ R z be linear operators such
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[December
that T; is bounded and onto and such that T1 = U1T;. Let Za be the (n a + 1)dimensional Banach space whose unit cell is T'aSx. The operator U a maps Szl onto T1Sx and by the choice of na there is subspace Y1 of Z1 of codimension 1 such that U1Sy~, contains all the extreme points of T1Sx and hence all of T1Sx. Let now el > 0 be such that Lemma 4.2 holds with 2el if TISx is taken as P. Choose next points ,fxl"lnl ~ ~~ = 1 in Sx and a 61 > 0 such that Y~x~ e Y1 for every i and such that whenever []c~-Tlx][]<=61for every i the convex hull C of {+c~}7=~1 satisfies t3C = ~l -nbdOT1Sx. (Remark: unless X is reflexive we do not have in general that T;Sx = T'ISx and hence we cannot insure the existence of points x~ ~ Sx such that Tax] ~ ext T~Sx. But it suffices for our purpose to choose the x~ so that the T~xl are near the extreme points of T1Sx). By Lemma 4.1 there is an operator 0 1 : R " ' + a ~ R 2 such that OlTIS x has 2n2 > 2nl extreme points, [l(.71T;x~-Tlx,'l[ < 6 j 2 for every / a n d Ot71T[Sx e 1 - nbdOT1S x. Put T2 = ~1T1'. Next we choose T~ : X ~ R "~+"~+ 1 and U2: R "~+"' + 1 ._, R2such that T2 = UzT~. We take Y2, a subspace of deficiency 1 in Z2 ( = R "~+"' + 1 with unit cell T2Sx) such that T~xt, e Y2 for every i and U2Sy, = U2Sz,. Let ez > 0 be such that Lemma 4.2 holds for 2e2, with P = T2Sx, and such that e 2 - n b d T 2 S x < e l - nbdOTaSx'. We choose 6 2 > 0 and t~'x2/"~ , ,~=1 ~ Sx as in the first step and then (by Lemma 4.1) we choose a 02 such that ]1UzTzxk--T2xk[] <6J22 for k = 1,2, and i = 1,... ,nk, OO2~2Sx=e2- nbdOT2Sx and such that the number 2na of extreme points of ~ 7 ~ f f x is greater than 2n2. Continuing we get sequences {nk}, {6k}, {ek}, {Tk} and (x k} such that (4.1) (4.2) (4.3)
nx
< n2 < "'" < nk < " " .
2n k is the number of extreme points of TRSx. ek-nbdaTkS k ~ ek_l-nbdOTk_lSk_l , ek ~, O.
(4.4)
For every symmetric convex body C in R 2 such that aC ~ 2ek-nbdOTkS k the number of extreme points of C is > 2n k.
(4.5)
t~if"k~"~si= t c S x and whenever llci - Tkx~[ I < 6 k f o r i = l , . . . , n 0 (convex hull of {_+ ci}~,k I) = ek-nbdOTkSk.
(4.6)
t[Tkx[- T~_lx[[]<~/2 ~, i=l,...,nj,
k we have
j = 1,2,...,k-1.
By (4.3) II Tkll < (1 + el)tl T1 !] for every k and hence by the w* compactness of the unit cell of X* the sequence (Tk} has a limit point Tin the strong ( = weak in our case) operator topology. By using (4.3) again we get that TSx c (1 + ek)TkSx for every k. Also, by (4.6), II Txk-- Tkxk I[ < 6k for every i and k and hence by (4.5) TSx = convex hull of { + Txk}7 k 1 = (1 -- ek)TkSx. We have thus that OTSx < 2e~-nbdOTkSx for every k and therefore, by (4.1) and (4.4), T S x is not a polygon. Q.E.D.
1966] ON KLEE'S PAPER "POLYHEDRAL SECTIONS OF CONVEX BODIES"
241
COROLLARY. No infinite-dimensional conjugate (and in particular reflexive)
Banach space is polyhedral. Section 5. Open problems and remarks. (a). The first problem we mention is one which was raised already in [4]. Does there exist a separable reflexive space which is universal for all finite-dimensional Banach spaces? Klee observed in [4] that the separable reflexive space ( 2~n%1 G l~)2 is universal for all polyhedral finite-dimensional spaces and hence, in an obvious sense, " a l m o s t " universal for all finite-dimensional spaces. It follows that in order to establish the nonexistence (or, of course, the existence) of such a universal space, non polyhedral finite-dimensional spaces must be considered. The method of Section 2 may be helpful in this direction. In fact, given any two-dimensional non-polyhedral Banach space B then, as is easily seen, there oo in B such that IIb, + b~lI ~ IIb, II + 11bell- 2 is an unbounded sequence { b ,},=1
for every i ¢ j . By Lemma 2.1 there is for every sequence of signs 0 = (01,02, 03,..') a three-dimensional space Co containing B and a vector uo ~ Co such that IIno I1-- 1 and IIno-O,b~ II--< 11b, 11- 1 for e v e r y / . It is easily verified that if for 0 ' ¢ 0" there are isometries T' and T" from Co, and Co, into a Banach space X such that T'[B = T"IB then II T ' n o , - z",+, II = 2 t h u s if x is separable there are for every isometry T: B ~ X at most a countable number of sequences 0 for which there is an isometry To:CoaX with TolB= T. It is hoped that an argument of this type suitably combined with a compactness argument will prove that there is no separable reflexive (or even conjugate) space which is universal for all three-dimensional spaces. It is conceivable that here there is a real difference between the two and three dimensional cases. (b) Can the method of Section 3 be modified so as to establish the nonexistence of universal spaces with respect to isomorphism and not only isometry? More specifically: let X be a Banach space, let 1 < p < oo and let M < oo. Assume that for every k there is an operator k
r~ :~x + .~.. + x L + x
Ilyll<=HT,yll
with <__Mlly[I for every y e ( X + . . . ~ ) X ) p . Must X have a subspace isomorphic to Ip if p < ~ or to Co if p = ~ ? Can one prove at least that X is not reflexive if p = 1 or ~ ? (c) Here are some problems concerning the existence of universal spaces in certain classes of Banach spaces. (i) Does there exist a separable strictly convex space which is universal for all strictly convex separable spaces? (ii) Does there exist a separable space with an unconditional basis which is universal (in the sense of isometry or isomorphism) for all separable Banach spaces with an unconditional basis?
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These questions are of the type appearing in Theorem 3.1, yet our method does not seem to provide any information concerning them. (d) Another kind of problem concerning infinite-dimensional universal spaces is exemplified by the following questions. (i) Does there exist a separable reflexive space which is universal for all Iv, 1
y, , max 2 y, 1
.
1_<~_~o~
It is easily verified that ext S x , consists of those y = (y~, Y2,'") for which l Y~] = ½ for two indices i and = 0 for all other indices. By using the simple criterion given in [7, p. 102] it follows now easily that X is polyhedral. Obviously S x has an infinite number of extreme points. Let us also mention the following fact. If X and Y are polyhedral spaces then (X @Y)I is also polyhedral. Since It is not polyhedral it follows by Lemma 3.1 that for every cardinal number Jr' there is no universal polyhedral space of density character Jr'. REFERENCES
1. C. Bessaga, A note on universal Banach spaces of finite dimension, Bull. Acad. Pol. Sci. 6 (1958), 97-101. 2. C. Bessaga and A. Petczyfiski, Some remarks on conjugate spaces containing subspaces isomorphic to Co, Bull. Aead. Pol. Sci. 6 (1958), 249-250. 3. M. I. Kadee, Linear dimension of the spaces L v and Lq, Uspehi Matern. Nauk. 13 (1958), 95-98. 4. V. Klee, Polyhedral sections of convex bodies, Aeta Math. 103 (1960), 243-267. 5. V. Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79-107. 6. V. Klee, On a conjecture of Lindenstrauss, Israel J. Math. 1 (1963), 1-4. 7. J. Lindenstrauss, Extension of compact operators, Mem.'Amer.tMath. Soc. 48 (1964). 8. L. Nachbin, A theorem of the Hahn-Banach type for linear transformation, Trans. Amer. Ma+h. Soe. 68 (1950), 28-46. 9. H. Nakano, Modularedsequence spaces, Proc. Japan Acad. 27 (1951), 508-512. Tn~ HEBREWUNIVERSITY o~ JERUSALEM
INTERIOR POINTS OF CONVEX HULLS BY
WILLIAM E. BONNICE AND JOHN R. REAY ABSTRACT
I f a set X c E n h a s non-empty k-dimensional interior, or if some point is k-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example if X c E n has intk X ~ 0, (0 _< k _< n) and if p e int con X, then p e i n t c o n Y for some Y ~ X with card Y_< 2 n - - k + l .
1. Results. For X in a linear space, the k-interior of X , denoted intkX, is the set of all points s such that s is in the relative interior o f some k-simplex contained in X ; equivalently s ~ intkX if and only if there exists a k-dimensional flat F such that s is interior to X n F relative to F . Note that into X = X and if X is a subset of an n-dimensional space then int, X is the usual interior of X , which will also be denoted by i n t X . We will let c o n X , a f f X , l i n X , and c a r d X denote the convex hull, the affine span, the linear span, and the cardinality of X , respectively. O f the following three results, the first two are due to Steinitz [5] and the third to Reay [4]. A. I f X c E " and p s i n t c o n X then p c i n t c o n Y for some Y c X with card Y < 2n. B. I f X c E" is not contained in the union of n lines through p and if p ~ int con X then p Eint con Y for some Y ~ X with card Y < 2n - 1. C. F o r n > 3, if X ~ E " is connected and if p ¢ X but p ~ i n t c o n X , then p ~ intcon Y for some Y ~ X with card Y < 2n - 2. The purpose of this paper is to prove (with other conditions on X) some analogous results, principally the following: 1.1 THEOREM. I f X c E " h a s intkX # S~5(0 < k < n) and if p e i n t c o n X p e i n t c o n Y for some Y ~ X with c a r d Y < 2 n - k + l .
then
1.2 THEOREM. For X ~ E" if there is a k-dimensional flat F (0 < k < n - l ) with int k ( X t7 F) # ~ (thus int k X # ~ ) , and if p q~F is such that p e i n t c o n X , then p ~ i n t c o n Y for some Y c X with card Y < 2n - k. 1.3 COROLLARY. [ f X c E"has i n t X # ~ and if p ~ intcon X then p ~ intcon Y f o r some Y = X with card Y = n + 1. N o t e that 1.3 is a special case of both 1.1 (with k = n) and of 1.2 (with Received October 26, 1966 243
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[December
k = n - 1). Except for the case k = 0 in 1.1, each of these is a best possible result in the sense that for each result there are configurations which satisfy the hypothesis, but for which card Y cannot be reduced further. 1.4 EXAMPLE In 1.1, to see that 2n - k + 1 is " b e s t " for 1 -< k < n let p be the origin in E", let B be a solid k-dimensional ball centered at p, let A be a linear basis for an (n-k)-dimensional linear subspace supplementary to linB ( A = ~ when k = n ) and let X = B u A u ( - A ) = { - a [ a e A } . Then if p e i n t c o n Ywith X = X , Ymust contain A U ( - A ) together with at least k + 1 points of B, so card Y > 2(n - k) + k + 1 = 2n - k + 1. 1.5 EXAMrLE. In 1.2, to see that 2n - k is " b e s t " for 0 < k < n - 1, let p be the origin in E", let q ¢ p, and let T be a solid k-dimensional ball centered at - q. with p ¢ aft T. (For k = 0, let T = { - q}.) Let V be a linear basis for an ( n - k - 1)-dimensional linear subspace supplementary to lin T = aff({p} u T) and let X = {q} u T u V u ( - V). Then if p e int con Y with Y c X , Y must contain V U ( - V ) U { q } together with at least k + 1 points of T, so c a r d Y > 2 ( n - k - 1) + k + 2 = 2 n - k. 2. Proofs. The proofs will be based on positive bases. Early papers on pos tive bases were those of Chandler Davis [2] and McKinney [3]. We will use the terminology and theory as presented in Bonnice-Klee [1, pp. 5-7], and Reay [4, pp. 5-8]. In brief, for a set U contained in a linear space L, pos U will denote the set of all finite linear combinations of elements of U having all coefficients nonnegative. If A is a subset of L such that pos U = A then we say that U positively spans A. Thus pos U is the cone or " w e d g e " generated by U having the origin as vertex. The set U is said to be positively independent if for all u e U, u pos (U ~ {u}). If U is positively independent and positively spans a linear space L, then U is a positive basis for L. Every linear space L admits a positive basis. In fact, if Lis n-dimensional and k is the cardinality of a positive basis for Lthen n + 1 < k < 2n and moreover all of these values of k are realizable. A basis with eardinality n + 1 is called a minimal basis for L. A linear subspace S o f L is a spanned subspace with respect to U if U n S positively spans S. In this case, if S is k-dimensional (k > 1) and U n S has k + 1 members (and hence U n S is a minimal basis for S), S is called a minimal subspace with respect to U. The important connecting link between positive bases and the results we want to derive about a point p ~ int con X is the obvious fact that for X contained in an n-dimensional space E, 0 e int con X if and only if pos X = E. 2.1 LEMMA. I f X positively spans an n-dimensional linear space E, if M is a d-dimensional (1 < d < n) subspace of E and if U ,-- X is such that pos U = M where card U =f, then there is a subset Y o f X with card Y<= 2(n - d) + f which positively spans E, and hence the origin is in int con Y. In particular if M is a minimal subspace with respect to X, card Y < 2n - d + 1.
1966]
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245
Proof. The case d = n has been noted above. For 1 -< d -< n - 1, let S be an (n - d)-dimensional subspace of E such that E is the direct linear sum of M and S and let 7r be the linear projection of E onto S with kernel M. Then zc(X-~ M) positively spans S and hence there is a subset W of X ~ M with n - d + 1 < card W__<2(n - d) such that zrW positively spans S. By assumption there is a U c X with card U = f such that pos U = M. Letting Y = U U W, it follows that pos Y = E (For details see [-1, L e m m a 2.7] or [4, L e m m a 2.3]) and that card Y_<_2(n - d) + f. Proof of 1.1 and 1.2. Since we may assume that p is the origin, 1.1 follows from 2.1 once it is shown that there is a minimal subspace with respect to X having dimension at least k. To do this, begin by taking a point q ~ intk X such that q ~ p and let B k be a closed solid k-dimensional ball centered at q and contained in X . By A. above, p is interior to some subset of X which has at most 2n points. Adjoining this finite set to B k we obtain a compact subset having p in its interior. Thus we may assume that X is compact and hence so is con X . Then the ray from q through p intersects the boundary of con X in some point r e con X . Let H be a hyperplane through r supporting con X . Since H n con X -- con(H n X by a theorem of Caratheodory ([4, Theorem 1.1]) there is a subset T of H n X with card T__< n such that r e con T. By taking card T minimal with respect to the property that r e con T, T may be assumed to consist of the verticles of a j-simplex (0 < j <_ n - 1) having r in its relative interior. Let L denote pos ({q} t3 T) and let F denote affB k, so that L A F is an affine space containing q. I f L n F = F then L = F and since q is in X , L is a minimal subspace with respect to X having sufficient dimension. I f L c3F is properly contained in F , let A be an affine subspace of F which would be supplementary to L n F in F if q were the origin. Letting e denote the dimension of A, then e > 1 and A n B k = B e is a closed e-dimensional ball centered at q. Thus there is an e-simplex with vertex set V contained in B e and having q in its relative interior. Then p o s ( V u T ) = lin(V U T) and d i m p o s ( V u T) = d i m a f f T + 1 = (card V) - 1 + card T so p o s ( V tA T) is a minimal subspace with respect to X which contains F (and p) and so is at least k-dimensional. This completes the p r o o f of 1.1. To prove 1.2, we note that under the stronger hypothesis of 1.2, B k may be chosen in the above p r o o f so that p ~ F = affB k and hence the ray from q through p intersects F only at q. Therefore aff((p) • F) is (k + 1)-dimensional. But minimal subspace p o s ( V W T ) contains aff({p} WF) and therefore is at least ( k + 1)-dimensional. N o w by 2.1 there is a subset Y of X with card Y < 2n - (k + 1) + 1 = 2n - k such that p e i n t c o n Y. 3. Apparent interiors. The contribution of a point x ~ X to the set p o s X is not determined by how far x is from the origin, but only by the direction of x
246
WILLIAM E. BONNICE AND JOHN R. REAY
December
from the origin. That is, x may be replaced in X by ~x (where ~ is any positive number) and pos X will not be changed. As a result conditions of the form " X is connected" or "intk X e Z ~ " may often be replaced by much weaker conditions in theorems where the proofs use the theory of positive bases. As an example, for a given point p ~ E", let ~rp denote the usual radial projection of E " ~ {p} onto the unit sphere centered at p. We say that X appears to be connected from p provided that ~p(X ~ {p}) is a connected subset of the sphere. The result C in the first section may now be given in the much stronger form: D. For n > 3, if X c E" appears to be connected from p and p e int con X then p e i n t c o n Y for some Y c X with card Y< 2n - 2. Note that the set X may appear connected from p and yet be totally disconnected. To obtain a similar generalization of 1.1 and 1.2, we say that X appears to have k-interior from p if there exists a k-dimensional flat F missing p and having the following property: If F' denotes the cone p + pos (F - p), (that is, the cone with vertex p and generated by F) and if ~p maps F' radially from p onto F then intknp(X U F') ¢ ~ . Equivalently, X appears to have k-interior from p if there exists a k-dimensional flat F missing p and such that in L = aff({p} u F) if F' denotes the open "half of L " containing F and bounded by the translation of F to p then the radial proection of X u F ' from p has nonempty k-interior. 3.1 TrIEOREM. I f X c E " appears to have k-interior from p(O <- k <- n - 1), and if p ~ i n t c o n X , then p ~ i n t c o n Y for some Y c X with card Y=< 2 n - k . Proof. We may assume that p is the origin. With the notation of the preceeding definition, let X'--7rp(X r3 p o s F ) c F and apply 1.2 to get a subset Y' of X w X ' for which p e i n t c o n Y ' with curdY' _-<2n-k. Now for each y ~ Y', • y y e X for some ~ y > 0 . Let Y = ( a r o ~ l y e Y ' }. Then Y c X , p ~ i n t c o n Y , and card Y< 2n - k. For 1 _ k -< n, we say that q e E" is k-surrounded by X c E" if there exists a k-dimensional flat F through q with the following property: If 7rq is the radial proiection of E" onto the unit n-shell {z e E : I z - q I = 1} centered at q, then ~ ( X n F) is all of the unit k-shell at q in F. (Note that if q is k-surrounded, then X appears to have (k-1)-interior from q.) With this notation we may obtain a theorem concerning any p e i n t c o n X whether X appears to have k-interior from p or not. 3.2 TI~Om~M. For X c E" if there exists some point of E "which is k-surrounded (1 <=k < n) by X and if p e i n t c o n X then p e i n t c o n Y for some Y c X with card Y = < 2 n - k + l . That 2 n - k + 1 is "best" is seen from Example 1.4.
1966]
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247
3.3 THEOREM. For X c E n and p ~ i n t c o n X i f there is a k-dimensional flat F missing p (1 -< k ~ n - 1) and a point q e F f o r which 7r~(X N F) is the entire unit k-shell at q in F (thus q is k-surrounded by X ) then p e intcon Y f o r some Y c X with c a r d Y =< 2 n - k + 1. Moreover, if there is a such a q in X then there is a Y as stated but having card Y=< 2n - k. That 2 n - k is best seen from Example 1.5. The following example shows that 2 n - k + 1 cannot be improved on. 3.4 EXAMPLE. Let V U W be a linear basis for E n where V and W are disjoint and Vhas k + 1 members. Let X = ( - W) u W u ( - V) u boundary con V. Then F = aft V is a k-dimensional flat and, if q is in the relative interior of k-simplex con V, 7r~(X u F) is the entire k-shell at q in F . Letting p be the origin, if p ~ int con Ywith Y c X , Ymust contain ( - W) u W u ( - V) together with at least two points of the boundary of con V. Thus card Y >__2(n - k - 1) + (k + 1) + 2 = 2n-k+l. Proof of 3.2 and 3.3. We may assume p is the origin. Since some q is k-surrounded by X , there is a k-dimensional flat F through q for which 7r~(X t~F) is all of the unit k-shall S at q in F . We may assume that q 4 p because 3.2 follows from 3.1 in the special case when p = q. The proofs proceed as the proofs of 1.1 and 1.2 except that the role of B k (the closed solid k-dimensional ball centered at q) is played by S. So in this setting, F denotes affS and again L denotes pos([q] U T) -- aff({p} U T). If q is in X or if L ~ F is a proper atfine subspace of F then as in that proof we obtain a subset V of S whose convex hull is an e-simplex having q in its relative interior, and V u T is a minimal positive basis for the subspace pos(V u T) of dimension at least k (dimension at least k + 1 with the hypothesis of 3.3). Now a point v e V might not be an element of X , but it is the radial projection gqx~ of some point x~ e X . If V' is the set of e + 1 points xv e X thus obtained from the e + 1 points of V, then pos (V' u T) = pos (V U T a n d the p r o o f is completed by applying 2.1 as before, obtaining Y c X with p e int con Y a n d card Y < 2 n - k + l in case of 3.2 and card Y < 2 n - k in the case of 3.3. There remains only the case where q ¢ X . In this case for both 3.2 and 3.3, we have to find a Y c X with p e intcon Y and card Y < 2n - k + 1. As noted above, we may assume that L n F = F and hence L ~ F . If p e F , then ray pq is contained in F and therefore intersects S. Thus there is a point x e X on the open ray from p through q. Hence pos({x} U T) equals L and is a minimal subspace with respect to X having dimension > k. Again 2.1 yields a desired Y. So assume that p ~ F . Any line in F through q has at least one point of X on each side of q. Let xl and xz be two such points of X such that q e c o n { x l , x 2 } . Then pos({xl,x2} u T) = pos({q} U T) = L and, since x 1 and x z are in
248
WILLIAM E. BONNICE AND JOHN R. REAY
F c L, pos ({xl,x2} UT) = L. With j + 1 denoting card T, since peL.,,F, j+l=dimL>dimF+l=k+l. Applying 2.1 with j + l playing the role of d and f = card({xl,x2} u T) (thus j + 1 < f < j + 3) there is a Y c X such that p ~ int con Y and with card
Y=< 2 [ n - (j + 1)] +f<= 2 n - j + 1 _< 2 n - k + 1.
REFERENCES 1. Wm. Bonnice and V. L. Klee, The generation of convex hulls, Math. Ann. 152 (1963), 1-29. 2. Chandler Davis, Theory ofpositive linear dependence, Amer. J. Math. 76 (1954), 733-746. 3. R. L. McKinney, Positive bases for linear spaces, Trans. Amer. Math. Soc. 103 (1962), 131-148. 4. J.R. Reay, Generalization of a theorem ofCarath~odory, Amer. Math. Soc. Memoir No. 54, 1965. 5. E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme I-II-IlI, J. Reine Angew. Math. 143 (1913), 128-175, 144 (1914), 1--40, 146 (1916), 1-52. UNIVERSITYOF NEW HAMPSHIRE, DURHAM, NEW HAMPSHIRE, MICHIGAN STATEUNIVERSITY, EAST LANSING, MICHIGAN AND WESTERN WASHINGTONSTATE COLLEGE
SOME MODEL THEORETIC RESULTS FOR co-LOGIC* BY
H. JEROME KEISLER ABSTRACT
The to-completeness theorem is applied to prove theorems above two-cardinal models, homogeneous models, and categoricity in power in to-logic. In this paper we shall apply the co-completeness theorem to obtain results about two-cardinal models, homogeneous models, and categoricity in power. By co-logic, L ~, we mean the logic formed by adding to first order logic L a new unary predicate symbol N and individual constants 0, 1, .... An co-model is a model for L'~ in which N = (0, 1,... }. The co-completeness theorem is a sufficient condition for a theory in co-logic to have an co-model; a statement o f it and the relevant references are given in §1. For an introduction, we shall give here a brief summary of our main results. Let K be the class of all co-models of a theory T in co-logic. If qS(x) is a formula and 92 a model for L ~', then q~(92) is the set of all elements of 92 which satisfy ~. In §2 we prove the following two-cardinal result: I f K has a model 92 of power m such that N O< I 1< m, then K has a model ~ of power N 1 such that I c~(~)l = No. The above theorem was proved by Vaught in [10] for the case where K is the class of all models of a theory in first order logic, and our result generalizes Vaught's theorem to co-logic. In §3 we use the results o f § 2 to prove: I f all models in K of power N 1 are homogeneous, then all uncountable models in K are homogeneous. This time the special case for first order logic appears to be new. (The definition and references for homogeneous models are given in §3). In §4 we apply the results of §3 to prove that: I f K has a homogeneous model of power N 1 and K is categorical in power N1, then K is categorical in every uncountable power. For a theory T in first order logic, Morley [8] proved that: if T is categorical in one uncountable power then T is categorical in every uncountable power. It is known from Morley and Vaught [10] that, assuming the continuum hypothesis, every first order theory T which has infinite models has a homogeneous model of power N1. Thus, if we assume the continuum hypothesis, we have a new (and much shorter) proof of the upward part of Morley's Received Aug. 20, 1966 * Our research was supported in part by NSF grants GP 4257 and GP 5913. 249
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theorem, and also a generalization of the upward part of Morley's theorem to co-logic. All our above results still work if K is a PC~' class, that is, the class of all reducts to L of co-models of a theory in c0-1ogic with countably many extra predicates. Besides the co-completeness theorem we shall make extensive use of the results of Tarski and Vaught [15], and Morley and Vaught [10]. Some applications of the w-completeness theorem to models of set theory will be given in [6]. The paper [7] contains theorems which are closely related to §3 and §4 of this paper, but do not use the co-completeness theorem. Most of our results in this paper were announced in the abstract [5]. 1. Preliminaries. We use the letters ~t,fl, ... for ordinals, and re, n, ... for cardinals. Cardinals are identified with initial ordinals. We shall work with a countable first-order predicate logic L with identity symbol. For the basic notions of model theory see, e.g. [13], [15]. Models are denoted by German capitals 92,~3,~, sometimes with subscripts, and the universe set of a model is denoted by the corresponding letters A, B, C. We shall sometimes enlarge the language L by adding new individual constant or predicate symbols. If a is an or-termed sequence of elements of a model 9.I for L, then (92, a) is a model of the language L(~) formed by adding 0t new individual constants to L. Similarly, ifR is an n-ary relation over A, then (92, R) is a model for the logic L(P) formed by adding a new n-ary predicate symbol P to L. The model (92,R) is called an expansion of 92 to a model for L(P), and 9.I is called the reduct of (92,R) to L. The symbols 9~ ----~, 92-<~ mean that 92 is elementarily equivalent to ~3, and 92 is an elementary submodel of ~3. An elementary chain is a sequence
92o "~ 92, "< "'"< 92. <'",
~
of elementary extensions. The fundamental result about elementary chains, due to Tarski and Vaught [15], is that the union U,<~92, of an elementary chain is an elementary extension of each model 92~. If T is a theory (set of sentences) in L, then a sentence ~b is a consequence of T if every model of T is a model of ~b. We say that ~ is consistent with T if T u {~b} has a model. If q~(x) is a formula of L whose only free variable is x, then we let ~b(92) = {a ~ A : a satisfies ~b(x) in 9i}. By co-logic we shall mean the language L ° obtained by adding to L a new
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unary predicate N and new individual constants 0,1,2, ..., one for each natural number. A model (%N,0,1,2,...) for L '~ is said to be an to-model if N = {0,1,2, ...}. Two simple remarks about to-models are:
Any submodel of an to-model is an to-model. The union of a chain of to-models is an to-model. A theory T in to-logic is said to be to-complete if (1) N(0), N(1), N(2),... are consequences of T, (2) If tk(x) is a formula and qS(0),q~(1), q5(2), ... are consequences of T, then Vx(N(x) ~ ~b(x)) is a consequence of T. The condition (2) is clearly equivalent to: (2') If q~(x) is a formula and 3x(N(x) A tb(x)) is consistent with T, then there exists n < to such that ~b(n) is consistent with T. The basic result about co-logic which we shall use is the to-COMPLETENESS THEOREM. Let T be a theory in co-logic. I f T has a model and is to-complete, then T has an co-model. Various forms of the co-completeness theorem were obtained independently by several people, including Schutte, Henkin, Orey, Ryll-Nardzewski, (see Addison [1], p. 36 for references). The above formulation is due to Orey [17]. A class K of models for L is said to be a pseudo-elementary class, or a PC~ class, if there is a finite or countable list of extra predicates Po, P1, ... and a theory T in L(Po,P:, ...) such that K is the class of all reducts to L of models of T. This notion is due to Tarski 1-13]. We shall say that K is an og-pseudoelementary class, or PC'~ class, if there is a finite or countable sequence of extra predicates Po,PI,"" and a theory T in L°'(Po, P~,...) such that K is the class of all reducts ~ to L of to-models (92(,N,O, 1, ...,Ro, R1, ...) of T. It is clear that if T is a theory in L then the class of all models of T is a PC~ class, but not conversely. Also, every PC~ class is a PC~'class. The notion of a PC~ class is more general than it might first appear. For instance, consider the infinitary logic Lo,,,~ which has all the symbols of L plus countably infinite conjuctions and disjunctions. Scott 1-12] has shown that if 0 is a sentence in L,o,~,, then the class K of all models of 0 is a PC'~ class. In particular, if T is a theory in L, ~ is a set of formulas a(x), and K is the class of all models ~ of T such that no element of 9.I satisfies all formulas of Y~, then K is a PC~ class. Likewise, if T is a set of sentences in the weak second order logic of Tarski [14], or a set of sentences in the logic with the extra quantifier "there exist infinitely m a n y " (see Fuhrken [3]), then the class K of all models of T is a PC'~ class. Finally, if L already has the symbols N, 0,1,..., then the class of all to-models of a theory T in L is a PC~'class.
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2. Liiwenheim Skolem theorems for two cardinals. Vaught in [10] proved the following theorem: Suppose 92 is a model for L, dp(x) is a formula, and No < [~(92)[ < Then there is a model ~ - 92 such that lqb(~)] = No and I c[ = N1.
IAI.
Chang conjectured that Vaught's theorem could be improved by replacing ~ - 9 / by E-< 92. Rowbottom [-11], however, showed that Chang's conjecture contradicts the axiom of constructibility. Our first theorem is an improvement of Vaught's theorem in the direction of Chang's conjecture. TIJEOREM 2.1. Suppose 92 is a model for L, qb(x) a formula of L, and No < Iq5(92)[ < A . Then there exist models ~ , ~ such that ~ <92.I, ~ "K~, ~(~) = ~b(~), ~b(~)] = No, and [ C[ = N 1. Proof. Let m = [ ~b(92)]. 92 has an elementary submodel of power m + which contains ~b(92). Thus we may as well assume that IAI = m + Let < be a well ordering of A of type m +. Let ~(xyvl... v,) be a formula in L(<) and a 1,'", a, ~ A. We note that in the model (92, <), if there are arbitrarily large y ~ A such that for some x ~ q5(92), ~(xyal ... a,) holds, then there is a fixed x ~ ~(92) such that for arbitrarily large y ~ A, ~b(xyal... a,) holds. In other words, the sentence below holds in (92, <): (1)
Vvx "'" v. [Vz3y3x(z < y/~ (a(x) /~ ~(xyvl... v.))
3xVz3y(z < y A dp(x) A t~(xyh'" v,))]. The main step in the proof is to show the following: (2) Every countable model (~o, < o ) = (92, <) has a countable proper elementary extension ( ~ i , < 1) such that ~b(~31)= ~03o). After (2) is established we may take a countable elementary submodel (~, <) of (9.I, <). Using (2) o~ times we construct an elementary chain <) = 030,
< ol,
such that for each a, [B~I =No, B ~ B ~ + t , and q~(~B~)= ~b(~). Let ¢ be the union of the elementary chain ~,, z < ol. Then ~B K ~, ~b(~) = ~b(~3), ] ~b(ff)[ = Ro, and ICI = N1. It remains to prove (2). First we make (~3o, < o) into an o-model (~o,
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~ * = ( ~ o ,
Vz3y(z < y A ~(Y))
(3) holds in f5*.
We also note that given any formula ~(Vo "") we may find a formula ff'(v o ... ), in which the symbols N,0,1,... do not appear such that V V o . . . ( f f ~ ' ) is a consequence of T. To form ~b' we replace N(x) everywhere by ¢(x) and replace numerical constants m by the constants cb which denote the same element of ~*. Suppose now that the sentence
3x(N(x) AO(xcbl "'" cb.c)) is consistent with T. We may assume that the symbols N,0,1,... do not occur in 0. By (3) the sentence
Vz3y3x(z < y A N(x) A O(XCbl"'"cb.Y)) holds in ~*. Then ~ * also satisfies the sentence
Vz3y3x(z < y A ¢(x) A O(xc~... cb,,y)). Using the fact that (~o,
~xVz3y(z < y A ¢(x) A O(xcb,... cb.y)) holds in ~*. Then for some m < o2,
Vz3y(z < y A O(mcb, ... cbnY)) holds in ~*. Using (3) again, we see that the sentence O(m,c~, .-. c~ c)
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is consistent with T. Therefore T is co-complete. By the co-completeness theorem T has an co-model (fB1,
Icl=N,.
Proof. For some theory T in an co-logic, L ~ (Po, P , ' " ) , K is the class of all reducts to L of co-models of T. We need only apply Theorem 2.1 to models of T with the formula ~b(x) = ~b(x) V N(x) in place of ~b(x). The point is that if ~* is an co-model of T,~3* -K ~*,~3' -KK*, and ~k(~3*) = ~b(G*),then ~3' and G* are also co-models of T. Morley in [9] proved a (quite different) two-cardinal theorem in which the formula t~(x) is an infinite conjuction of formulas of Linstead of a single formula. Our next corollary is a generalization of Vaught's two-cardinal theorem in which ~(x) can be a formula in L~,,~ (this includes as a special case the infinite conjuctions @(x) of formulas of L). COROLLARY 2.3. Let ~ be a model for L,~(x) a formula in L,oio, and No <=]dp(~)] < A[. Then there exist models ~ , ~ such that f~ -K~,fB -K~, d~(~) q~(~, [q~(~) = No, and [C 1 = N1. Moreover, if K is a PC~class and ~ K , then we may also choose ~ , ~ so that ~ e K , ~ e K . Proof. Add an extra unary predicate symbol U to L. Let K' be the class of all models (~3, V) for L(U) such that ~ - 9.I and V = 4~(~) (and, if a K is given, ~3 e K). V = ~(~3) is true if and only if (~3, V) satisfies the sentence
Vx( u (x)
¢(x))
of L,,,,,. It follows that K' is PC~. Now apply Corollary 2.2 to the class K ' with U (x) in place of ~b(x). Corollary 2.2 is not in general true if we allow classes K which are defined using theories in a logic with uncountably many extra predicates. For instance, we could take the K which consists of those models 9.I for which ]~b(A)[ > N 1. Moreover, using an example is Scott [12] one can easily find an uncountable set 12 of sentences in L,ol, such that Corollary 2.2 is false for the class K of all models of 12. As a last corollary, we apply the proof of Theorem 2.1 to give a necessary and sufficient condition for a theory in co-logic to have an co-model of power N1.
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COROLLARY 2.4. Let T 1 be the theory in the logic L c° ( < ) consisting of the following sentences: " < is a strict linear ordering"; Vx3y(x < y); every sentence of the form Vvl ... Vv,[Vz3y3x(z < y A N(x) A ~(xyvl "'" v,)) --* 3xVz3y(z < y A N(x) A ~(xyvl"'" v,))]. A theory T in L '° has an o~-model of power N 1 if and only if the theory TL) T1 in L°~ ( < ) has an m-model.
Proof. As in the proof of Theorem 2.1, one can show that every countable m-model of T U T1 has a proper elementary extension which is an ~o-model. It follows that T U T1 has an o~-model of power N1, and the reduct of this model to L '° is an o-model of T of power Nx. Furken [3] gave an improvement of Vaught's theorem in which the formula ¢(x,y) is allowed to have the extra parameter y. We shall extend Fuhrken's theorem to a~-logic. We need the following well-known generalization of the o~-completeness theorem: I f T is consistent and w-complete with respect to (Ni, Oi, li,"" )for i = 0,1,2, ..., then Thas a model which is an o-model with respect to each (Ni, Oi, li,... ). If b e A, and ~(x, y) is a formula, we shall let ~(9.I, b) = {a e A: (a, b) satisfies ¢(x, y) in ~}. THEOREM 2.5. Suppose K is a PC'~ class, dR(x,y) is a formula of L (or of L~lo) , and 9.I is a model in K whose power IAI is regular and such that
(i)
[¢(9.I,b)[ < [A[ for all b e A .
Then K contains a model ~ of power N 1 such that
(ii)
I ¢(~' c)] __
Proof. It suffices to prove that for every o~-model 9.I of regular power and with the property (i), there is an ~o-model ~ - 9.:[ of power Nx with the property (ii). We argue as in the proof of Theorem 2.1. Let < be a well ordering of A of type [A I" Since JAils regular, the model (9~, < ) satisfies the formula (1) with ¢(x, vl) in place of ~b(x). Let (~o, < o) - (~, < ) be a countable w-model. For each b e B o, let Nb = ¢(~o, b) = {Oh,lb, "'" }.
Using the co-completeness theorem with respect N and N b for each b ~ Bo, we see that (~o,
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t~(~0, b ) = ~b(~l,b ) for all beBo. By iterating this construction co1 times we obtain the desired model ~. Notice that Theorem 2.5 is a corollary of Theorem 2.1 when the power of 92 is a successor cardinal, but not when the power of 92 is an inaccessible cardinal. Fuhrken 13] gave an example showing that his theorem fails when the power of 92 is singular, and it follows that our Theorem 2.5 also fails in that case. 3. Homogeneous theories. Consider a model 92 for L of uncountable power m. We say that 92 is homogeneous if for any two elementary submodels ~3, ~ -< 92 of power less than m, any isomorphism of ~3 onto ~ can be extended to an automorphism of 96. This notion is due to Morley and Vaught [-10] and is based on earlier work of J6nsson [-4]. A more useful equivalent definition of homogeneous is given in the following/emma of Morley and Vaught [10]. LEMMA 3.1. A model 92 for L of power rrt is homogeneous if and only if the following holds. For all a < m and all a, b E A ~, if (92, a) = (9.I, b)
then for all c EA there exists d c A such that (92,a,c)=(92,b,d). The next lemma is an observation of Morley. LEMMA 3.2. Suppose 92 and ~ are homogeneous. Then 92 and ~ are isomorphic if and only if
(1) IA[=I I; (2) For every finite sequence a in A there is a finite sequence b in B such that
(92, a) - (~3, b), and vice versa. Proof.
I f 92 and ~ are isomorphic then (1) and (2) obviously hold.
Assume (1) and (2). Using Lemma 3.1, one can show by induction on a that for each ~ < m and a ~ A" there exists b e B" such that (9~, a) --- (~, b), and vice versa. By a "back and forth" argument it follows that 92 and ~3 are isomorphic. We now prove the main theorem of this section. THEOREM 3.3. Suppose that K is a PC~' class. I f every model 9AeK of power N~ is homogeneous, then every uncountable model in K is homogeneous. Proof. Let K contain a model 92 of power m > N1 which is not homogeneous. We shall find a model ~ e K of power ~1 which is not homogeneous. By Lemma 3.1 there exists 0c< m, a, b e A ~, and c s A such that (92, a) = (92, b) but there is no
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d ~ A with
257
( ~ , a , c ) - ( q I , b,d).
Now let P be a new binary predicate, let
R = {(~p, bp):fl < a}, let q~(x) be the formula
3yP(xy) ~/ 3yP(yx), and consider the model (9.I,R). We have q~((~, R)) -- range(a) k9 range (b), and hence
1~((9~, R)) I < m. Let
~o(Vo), ~dVoVl),
%(VoVlVO, ""
be a list of all formulas of L. The list is chosen in such a way that all the free variables of ~b, are among vo,..., v,. Then (gA,R) satisfies the following sentence 0 of L~,,~(P):
3xoVYo V 3x1"'" x~yl"'" y. n<¢9
[P(xlYa) A "" A P(XnYn) A -7 (~.(Xo "'" x,) ~-~tp.(yo "'" y,))]. Let K ' be the class of all models 03, S) of 0 such that ~ e K. Then K ' is PC~ and (9~, R) e K'. By Corollary 2.2 there is a model (if, S) e K ' such that (E, S) - (9,I, R), I ~b((E, S)) I = No, and [ C I = Na. Let (a~b~), (a~b~),... be a list of all pairs (c, d) e S. The list is countable because [ ¢((~, S))] = N o. It follows from (E, S) _=(9/, R) that (¢, a ') = (~, b'). Since (~, S) satisfies 0, there is a c' e C such that for no d' e C does
(¢,a',c')-(~,b',d'). Hence ff is a model in K of power N 1 which is not homogeneous. Theorem 3.3 above seems to give a new result even for first order logic: COROLLARY3.4. If T is a theory in L and every model of T of power N~ is homogeneous then every uncountable model o] T is homogeneous. In Theorem 3.3, or even in the special case 3.4, we cannot replace N~ by an arbitrary uncountable cardinal. For Morley has given an ingenious example of a theory T in L such that all models of T of power at least 2 2% are homogeneous but T has models of each power less than 2 2% which are not homogeneous.
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4. Categorical theories. In this section we shall give a relatively short new proof of the upward part of Morley's categoricity theorem (using the continuum hypothesis). Actually we shall prove a more general result in Theorem 4.1, involving PC~ classes. Theorem 4.1 does not use the continuum h y p o t h e s i s - it comes in only when we derive the upward part of Morley's theorem in Corollary 4.2. We shall say that a class K of models is categorical in power m if all models in K of power m are isomorphic. We allow the possibility that K has no members of power m. THEOREM 4.1. Suppose K is a PC'~class, K is categorical in power N1 and K contains a homogeneous model of power N 1. Then K is categorical in every uncountable power. Proof. By Theorem 3.4, all uncountable models in K are homogeneous. Suppose 91, ~ e K , both 91, ~3 have power m _->N1, and 91,~3 are not isomorphic. Then by Lemma 3.2 there is, say, a finite sequence al,...,a, e A such that for no b l , " ' , bn e B do we have (91, al, ..., a,) - (~, b~,..., b,). Choose 91o "~91, ~3o -<~3 such that 91o,~3oeK, IAol = IBo[ = N,, and ax, ..., a, e A o. Then Bo has no finite sequence b~, ..., b, such that (91o, al, "-, a,) - (~o, bD "", b,). Hence 9Io and ~o cannot be isomorphic. This contradicts our assumption that K is categorical in power N1. Therefore K is categorical in all powers m _->Na. COROLLARY 4.2. Assume the continuum hypothesis 2~° = N1. Suppose K is a PC~ class and K is categorical in power N 1. Then K is categorical in every uncountable power. Proof. It is shown in Morley and Vaught [10] that, if 2 ~° = N1, then every PC~ class which has infinite models has a homogeneous model of power Na. The author first proved 4.1 in a way similar to the proof of 3.3 without using Lemma 3.2. We are indebted to Morley for pointing out that the proof could be simplified by using Lemma 3.2. For some examples of theories T in L which are categorical in power Nx see Morley [-8]. In each known example, it happens to be obvious that T is categorical in every power m > N~, because the proof that T is categorical in power N1 also works for all m > N~. The same thing happens for each known example of a PC'~ class K which is categorical and has a homogeneous model of power Nx. The following are examples of PC6 classes K which are categorical in power N1 but cannot be characterized as the class of all models of some theory T in L.
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1. The class of all models (A,E) where E is an equivalence relation over A, each equivalence class has power [A I, and there are [ a l different equivalence classes. 2. The class of all models (A, El, E2) where El, E 2 are both equivalence relations with I A I equivalence classes and each equivalence class of E 1 meets each equivalence of E 2 in exactly one element. 3. The class of models (A, Po, P1,P2,'") where Po, P1,P2, ... are disjoint subsets of A of power and
IAI,
IA-UoP, I=IAI • 4. The class of abelian groups (A, + ) which have [A [ elements of order 2, order 3, and order 6, and no elements of any other order. 5. Let L' be an extension of the language L, let T be a theory in L' categorical in power N~, and let K b e the class of all reducts to L of models of T. We shall now list some examples of PC'~classes to which Theorem 4.1 applies. We shall give examples of PC~ classes K with the following property: (*) There is a countable extension L' of the logic L and a PC~' class K' of models of L' such that K' is categorical and has a homogeneous model in power N~, and K is the class of all reducts to L of models in K'. By Theorem 4.1, the class K' in the condition (*) is categorical in every uncountable power. It follows at once that every class K with the property (*) is a PC'~ class categorical in every uncountable power. 7. For each equational class M of algebras, the class K of all free algebras in M, also the class K' of all models (92, U) where 92 is an algebra in M freely generated by U. Note that K' has a homogeneous model of power K 1. 8. The class K of all models (A, E) where E is an equivalence relation over A all of whose equivalence classes are of power No. A suitable class K' is the class of all co-models (A,E,F, N, ...) such that E,F are equivalence relations over A, N is one equivalence class of E, and each equivalence class of F meets each equivalence class of E in exactly one point. 9. Like 8 but all equivalence classes of E have finite power, and for each n < to there are I A [ classes of power n. 10. Like 8 but all equivalence classes of E have power and there are No equivalence classes. 11. The class of all abelian groups (A, + ) in which the order of each element is a product of distinct primes and for each prime p there are elements of order p. 12. The class of all trees (A, < ) in which each element has finitely many predecessors and [ A [ immediate successors (and the tree has only one root).
IAI
IAI
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13. The class of all models isomorphic to a model (A, _~) where A is the set of all finite subsets of a set X. 14. The class of all models (A, Po, P1,...> where the P / a r e disjoint subsets of A of power [A[ and U i < t a P i = A. 15. Let X be a countable set of subsets of c~. The class of all models (A, Po, P i, ...> where each P~ is a subset of A, and for each y coo,
['~ Pi (3 0 (A - Pi) ey
i~y
has power t A [ if y ~ X and is empty if y ¢ X. 16. Let F be a countable field. The class of all pure transcendental extensions of F. 17. Let 9~ be a countable algebra which has an element 0 idempotent for all the operations of 9/. The class of all weak direct powers 9~~ (whose universe set is the set of all functions f e A 1 with f(i) = 0 for all but finitely i e I). Note that 4, 1 are special cases of 17. 18. Let ~ be a countable model for L. The class of all cardinal multiples 911 of 9~ (unions of disjoint copies 9~i, i e I, of 91). 19. Let Ko, K1,'" all have the property (.), and let K be the class of all cardinal sums 9/0 + 9g1 + ... of models 9/n e Kn all of the same power. 20. Let K o have property (.) and let K be the class of cardinal multiples 911, where 9~e K0 and :-II1. The following problem is open: Generalize the results of this paper by replacing N~ by an arbitrary uncountable cardinal. For example, are the following three things true for all m > N1 and all PC~ classes K, or if not, what else must be assumed about K or m? A? If there is an 9~ e K with
IAI
m = l (9 )l < tal, and if N 1 < n < m, there is a ~ e K such that
.-- 1+( 3)1 <181 . B? If every model of K of power m is homogeneous, then every model of K of power greater than m is homogeneous. C? If K is categorical in power m then K is categorical in every power greater than m. D? Suppose that T is a complete theory in L and every model of T of power N 1 is homogeneous. Does it follow that T is categorical in power Nl?
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The referee has pointed out that the answer to question A is negative when m = N,~ (assuming the G C H ) , because o f an example due to Chang. By c o m bining C h a n g ' s example with a result o f Morley [9], we see that question A has a negative answer whenever m is a singular cardinal o f cofinality less than N,~I(GCH ). To make question A reasonable, we must stipulate either that m is regular or that m has cofinality at least N,~. REFERENCES 1. J. W. Addison, The theory of hierarchies, in Logic, Methodology, and Philosophy of Science, Eds. Nagel, Suppes, and Tarski, Stanford, 1962, 26-37. 2. C. C. Chang, Ultraproducts and other methods of constructing models, to appear in Sets, Models mand Recursion Theory. ed. John Crossley, North Holland. 3. G. Fuhrken, Languages with the quantifier "there exist at least X~". in The Theory of Models, eds. J. Addison, L. Henkin, and A. Tarski, Amsterdam, 1965, pp. 121-131. 4. B. J6nsson, Homogeneous universal relational systems, Math. Scand. g (1960), 137-142. 5. H. J. Keisler, Homogeneous theories (abstract), Notices Amer. Math. Soc. 12 (1965), 600. 6. H. J. Keisler and M. Morley, Elementary extensions of models of set theory. To appear. 7. - On the number of homogeneous models of a given power. To appear. 8. M. Morley, Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), 514-538. 9. - Omitting classes of elements, in The Theory of Models, eds. J. Addison, L. Henkin and A. Tarski, Amsterdam, 1965, pp. 265-273. 10. M. Morley and R. L. Vaught, Homogeneous universal models, Math. Scand. 11 (1962), 37-57. 11. F. Rowbottom, Some strong axioms of infinity incompatible with the axiom of constructibility. Doctoral dissertation, Univ. of Wisconsin, Madison, 1964. 12. D. Scott, Logic with denumerably long formulas and finite strings of quantifiers, in The Theory of Models, eds. J. Addison, L. Henkin, and A. Tarski, Amsterdam, 1965, pp. 329-241. 13. A. Tarski, Contributions to the theory of models, I and H, Indag. Math. 16 (1954), 572-588., 14. ~ Some model-theoretical results concerning weak second order logic (abstract), Amer. Math. Soc. Notices, 5 (1958), 673. 15. A. Tarski, and R. L. Vaught, Arithmetical extensions of relational systems, Comp. Math. 13 (1957), 81-102. 16. R. L. Vaught, Models of complete theories, Bull. Amer. Math. Soc. 69 (1963), 299-313. 17. S. Orey, On to-consistency and related properties, J. Symb. Logic, 21 (1956), 246-252. UNIVERSITY OF WISCONSIN, MADISON, WISCONSIN
ON EXTREME POINTS IN SEPARABLE CONJUGATE SPACES BY
C. BESSAGA AND A. PELCZYI~ISKI ABSTRACT
It is proved that every bounded closed and convexsubset of an arbitrary conjugate separable Banach space is the closed convex hull of its extreme points. By the classical Krein Milman theorem, every convex bounded and weak-star closed subset of a conjugate Banach space is the weak-star closed convex hull of its extreme points. In general, the assumption of weak-star closedness cannot be replaced by norm closedness. For instance the unit ball of co is a closed bounded subset of m = l* and has no extreme points. However in the separable case we have: THEOREM 1. Every bounded closed and convex subset of an arbitrary conjugate separable Banach space X is the closed convex hull of its extreme points.* This theorem gives a useful criterion for a Banach space of being not isomorphically embeddable in any separable conjugate Banach space. COROLLARY. The space L(O, 1)is not isomorphic with any subspace of any separable conjugate Banach space (cf. Gelfand [2], Petczyfiski [6]). This follows from the fact that the unit cell in L(0, 1) has no extreme points. The assertion of Theorem 1 in the case X = l has been recently obtained by Lindenstrauss [5]. The proof of Theorem 1 is a slight modification of Lindenstrauss's proof. The special properties of l are replaced by the properties of X expressed in terms of inclinations d, (used by Kadec for constructing a homeomorphism between separable conjugate spaces). According to [5, Lemma 1] the Theorem 1 is reduced to the following PROPOSITION. I f K is a bounded closed convex subset of a separable conjugate Banach space, X then the set e x t K of all extreme points in K is non empty. Proof. Assume that I[" f] is an admissible norm of X such that for the sequences in the unit sphere {x ~ X ; I [ x H= 1} the weak-star convergence coincides with the norm convergence (such a norm exists, see Kadec [3] and Klee [4]). Let X = Z* and let (z,) be a linearly independent and linearly dense sequence in Z. Received November 29, 1966. * Since every complex-conjugatespace, regarded as a real space is a closed subspace of the suitable read conjugate space, it is enough to restrict the exttention to the real case. 262
1966]
ON EXTREME POINTS IN SEPARABLE CONJUGATE SPACES
263
Let L. = {x ~ X: x(zi) = 0 for i < n}. Then (L.) is a decreasing sequence of linear subspaces of X, and ['].oo= 1 L . = {0}. For any x in X let
d.(x) = inf IIx - u [I
for n -- 1,2, ...
u~.L..
We shall also consider the space l of absolutely summable real-valued sequences with the usual norm. For any ¢ = (¢k) ~ l let us set
d,(O=
~ [¢k[
for n = 1,2,...
k=l
Let V be either X or l~ Denote
B v -- the unit cell in space V,
T:(O
=
d.(x)= Ilxll-
,
for e > O
LEMMA 1. (Kadec [3], Klee [4]). There is a homeomorphism h: X
onto
~1
such that d , ( x ) = d,(hx) for n = 1,2,.... This homeomorphism has obviously the property: h(B x) = B' and h(TX(e))= Tt,(e). Since the set Tt,(e) is contained in the e-neighbourhood of the compact set B zc3 {4 e l: ~k = 0 for k > n}, we get LEMMA 2. The set Tt,(e) admits a finite 2e-net. LEMMA 3. I f (pn) is a sequence of positive integers ,(F,) is a decreasing sequence of closed sets with F k c TVk(1/k), k = 1,2,... and V is either l or X, then the set F = N°~= 1 f k is non empty and compact. Proof. 1° Let V= I. Let x, e Fn. By Lemma 2, (x,) is totally bounded, hence it has a cluster point Xo = 0,~°-- 1F,, i.e. F # ~ . From Lemma 2 it also follows that F is totally bounded, and since F is closed, it must be compact. 2 ° In the case V = X the assertion follows from Lemma 1 and from 1°. LEMMA 4. I f K is a closed convex subset of X with supx ~K I1x I[ = M < 1, then for every ~ > 0 there exists a closed face F 1 in K and a positive integer n such that F1 ~ TX(e). Proof. Take an y in K such that I]Yll > = M - e]4. Let n be such that d,(y) >_-M - e/2. Let f be a linear functional such that f ( x ) < d,(x) for all x ~ X and f ( y ) = d,(y) ( f is a supporting functional of the "cylinder" {x:d,(x) 6 1} at the point y/d,(y). By the Bishop Phelps theorem [1], there exists a g ~ X* with I I g - f l l < e / 4 M such that the face F~={xeK:g(x)= sup,~x g(u)} is nonempty. For x E F t we have
264
C. BESSAGA AND A. PELCZYI(ISKI
>__s(x) ->__
- Ils- g II !1 II ->- g(Y) -- lif- g I111x
>f(s)-llf-gll(ll~ll+jlylJ) =
>M =
~ 2
[I
~>]lxll 2
=
-~.
Hence F1 c TX(e). Q.E.D. Proof of the proposition. Without loss of generality we may assume that K c B x. By L e m m a 4, there is a sequence (Fn) of closed faces of K and a sequence
of integers (pn) such that Fn+ 1 is a face o f F , and F~ c TX(1/n). Hence, by L e m m a 3 F = ( ' ~ o~= 1 F , is a nonempty compact face of K. By the Krein Milman theorem, ext F # ~ and hence ext K # ~ . This concludes the proof. PROBLEM. Let X be a (separable) Banach space with the property that every bounded closed convex subset of X is the closed convex hull of its extreme points. Must X be isomorphic with a closed linear subspace of a (separable) conjugate space?
REFERENCES 1. E. Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Symposia in Pure Math. VII (Convexity), Amer. Math. Soc. 00 (1963), 27-37. 2. I. M. Gelfand, Abstrakte Funktionen und lineare Operatoren, Mat. Sb. 4 (46) (1938), 235-286. 3. M. I. Kadec, On connection between weak and strong convergence (Ukrainian), Dopovidi Akad. Nauk Ukrain. R.S.R. 9 (1959), 456-468. 4. V. L. Klee, Mapping into normedlinear spaces, Fund. Math. 49 (1960), 25-34. 5. J. Lindenstrauss, On extreme points in 11, Israel J. Math. 4 (1966), 59-61. 6. A. Petczyfiski, On the impossibility of embedding of the space L in certain Banaeh spaces, Colloq. Math. 8 (1961), 199-203. UNIVERSITYOF WARSAW, WARSAW,POLAND
O N PERFECTLY H O M O G E N E O U S BASES IN B A N A C H SPACES BY
M. ZIPPIN* ABSTRACT
It is proved that a Banach space is isomorphic to co or to lp if and only if it co has a normalized basis {X i}i=l which is equivalent to every normalized co 1" block-basis with respect to {X i}i= 1. Introduction. F. Bohnenblust gave in [2] an axiomatic characterization of c o and Ip. The following proposition follows easily f r o m his proof: oo PROPOSITION 1.1. Let {X i}/=1 be a normalized basis in a Banach space X . I f f o r every normalized block-basis { Y , } i ~ (with respect to {xi},~l) and any a oo real sequence { i}i=1
i=1
a:,lI =
"=
m f o r all natural n then the basis {X i}i=l is equivalent to the unit-vectors basis in c o or in lp f o r some p > 1. Moreover, this equivalence of the bases induces an isometric isomorphism of X onto c o or Ip. T h e following natural question arises: I f we assume only that all normalized block-bases with respect to { x ~ } ~ are equivalent, is {x~}~=~ equivalent to the unit-vectors basis in Co or Ip? In this p a p e r we show that the answer is positive and the equivalence of all normalized block-bases characterizes the unit-vectors bases in c o and l~. After the preliminary l e m m a s of Section 2, we use the m e t h o d of the p r o o f o f L e m m a 4.3 o f [2] to prove our main result, T h e o r e m 3.1. A r e m a r k concerning a result of A. Petczyfiski and I. Singer [6] concludes Section 3.
DEFINmONS AND NOTATIONS. A basis {xi}~= ~ in a Banach space is called normalized if x, = 1 for every i. The sequence {Yi}i:l is called a block-basis + 1) ~ajxj, where {p( )}i=~ with respect to the basis {x~}i~o = ~ if for every i Yi = ~v ,j ( i=pci)+
II II
Received December 21, 1966. * This is part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful guidance and for the interest he showed in the paper, and the referee for his valuable remarks. 265
266
M. ZIPPIN
[December
is an increasing sequence of nonnegative integers. In this paper we discuss only one basis {X i}~oo= 1 in the Banach space X ; all block-bases mentioned are assumed to be block-bases with respect to the basis {xi}~= 1. A basis {xi}~°:1 in X is equia oo valent to a basis {z i}~=1 ~ in a Banach space Z if for every real sequence { i)i=t ~i ~=1 aixi converges if and only if ~ t aiz~ converges. The closed subspace oo t in X is denoted by [yj~O . A sequence which is spanned by a sequence {Y..}~= {y,}~= ~ in X is called a basic sequence if it forms a basis in [yi]~o=;. (Every blockbasis is a basic sequence in X.) Following C. Bessaga and A. Petczyfiski [11 we call a basis {xi}~ 1 perfectly homogeneous if it is normalized and every normalized block-basis {z~}~°:t with respect to {xi}~°=l is equivalent to the basis {xi}~°:~. 2. Preliminary lemmas. Let {xi}i~=t be a normalized basis in a Banach space X and let {f~}~; denote its biorthogonal sequence in X*. In the sequel we shall consider the following property: (a)
If S and T are disjoint finite sets of positive integers and E a i x i + t " ~, aix i teS
>=
leT
~, aix t + s ieS
Itl ___lsithen
~, aix i ieT
for every real {a~}, i ~ S U T. LEMMA 2.1. Proof.
I f a basis {x i}i=1 ~ s a t i s a e s (a) then
II/,ll >--Y,(~,)=
liT, II =
l for
every i.
1. On the other hand /co
,I,,il-- o up
Ilj=l'~xJll-~l
,~=,
\
o xj _- o sup ,
II~J=, ,,x~ll~l
l a, l
since, by (a), la, I = tl a,x, ll = II ET=x aJx~ll z 1. X
co
LI~MMA 2.2 Assume that { ~}~=t is a basis in a Banach space X which satisfies n (a). I f [ s, [ < It, [ for 1 <<-i < n then tI ~,i=t s,X i II < 1t Z:=I tix, I[" Proof.
Use (a) n times.
LEMMA 2.3. Let {x~}i=l oo be a normalized basis in a Banach space X which oo satisfies (a). I f for some M > 1 ~,,~lx, II<:M for every n then {X ,},=1 is equivalent to the unit vectors basis of co.
11
Proof.
By (a), Lemma 2.1 and Lemma 2.2
max
l
i= 1
(max In,I) II 1,i~n
Hence, ]~o= 1 aixi converges if and only if as --* 0. f--*o0
i= 1
max a,I
1 ~f~n
1966]
PERFECTLY HOMOGENEOUS
BASES I N B A N A C H SPACES
267
~o be a perfectly homogeneous basis in a Banach LEMMA 2.4. Let {x i}i=1 space X. Then {xi}~oo= 1 is an unconditional basis. Proof. Since for any sequence {ai}i= oo t where a i = +_ 1 the sequence {aixi} ~o i= 1 is a normalized block-basis, ~ 1 a ibix~ converges if and only if ~ i ~ 1 bixi converges. Hence, ~ 1 bix~ converges unconditionally whenever it converges. This proves Lemma 2.4. X oo Assume, now, that { ~}~=1 is a perfectly homogeneous basis in a Banach space X . By Lemma 2.4 {x~}~ 1 is an unconditional basis. Assume, further, that {xi}~= t satisfies (a). Denote by {{y~}i~ t}~z the set of all normalized blockoo ~, I being the suitable index set. The asbases with respect to the basis {X ~}~= sumed equivalence of the bases induces, for each a e I, an isomorphism T~ from X o n t o [ Y i ] i =ool , defined by T~(~,~°=la~xi)= ~°=la~y~.(This follows fromthe closed graph theorem.) L~MMA 2.5.
There exists a real M >=1 such that for every a e I both HT, I1<=M
and IIr,-' il <=M Proof. Let us first show the existence of a finite bound for the set {11T~[l:a e I}. If {ll T, ll:aeI} is not bounded, then by the theorem of Banach and Steinhaus there is an x = ~ i ~ t b , x t 6 X such that and the set {llT~xl[:c(eI} s not bounded. Hence, we can select a sequence {~t(n)}~=l c l so that for every n z , ~ ---- n + 1. We construct inductively three sequences of positive integers {n(i)}, {p(i)} and {q(i)} in the following way: n(1) = 1, p(1) = 1 and q(1) is so large that [1 vq(1) ,,~(.(1)) I[ > 1. Suppose that n(1), n(2), ..., n(k), ,.,i =p(1) I.~'iy~ p(1),p(2),...,p(k) and q(1),q(2),...,q(k) were chosen such that
llx[I--1
II
(2.1)
b,YT(')l[
qz(i) biY~ ('(~))
> 1 for l ~ j < k
i = p(j)
(2.2)
q(j - 1) < p(j) <-_ q(j) for 2 < j =< k
(2.3)
If Mj (respectively, N j) is the least (respectively, the largest) index of the xis' which appear in the representations of ya(a(j)) ,,a(n(j)) then N j < M j + 1 p(j) , Y,,a(a(j)) p ( j ) + l , " " , Yq(j) for l _ = j = < k - 1 .
Choose n(k + 1) > max {Nk, q(k)} + 3, and put p(k + 1) = max {Nk, q(k)} + 1. By ( a ) a n d Lemma 2.1, for j >_-1 [bj[=[fj( ]~=lbiXl)[ <=[[ x N = 1, and since l[ y~(n(k+l)) H = 1 it follows that II--J'~"~P(k+ 1)-1/~= 1 ujyj"~("(k+a))ll~"~ V~'~'t1'"1"x).'' Therefore o b j.,vj • ~(.(~+1)) >.~. {1 ~ z,.aj=p(k+l)
n(k + 1) - p(k + 1) >= 2.
Choose q(k + 1) so large that l] ~.~j=p(k+l)ujyj vq(k+ 1) t..~(.(k+ 1)) 11> 1. Since the representation of each block y~(~(k+1))contains at least one xi, the choice of p(k + 1)
268
M. ZIPPIN
[December
ensures that (2.3) is satisfied for j = k. By (2.3) the sequence {y7("(k)) }v~k)~,s~(k).k~ 1 forms a normalized block-basis. By (2.1) Ek=l(,.q=v(k)°° S'q(k) b~yz"~"(k)h not converge while ]~o= 1(~~q(k) K ~" "~ converges, since, by Lemma 2.3, {X ~}~=t oo ,-,g=p(k)'q*V is an unconditional basis. This contradicts the equivalence of the block-bases. Assume that the set {ll T~xI[ : ~ I} is not bounded; so there exist sequences oo n {~(n)}~=~ = I and {Z ,}.=1 = X such that if z, = Z~o=~bjxj then for every n V oo b.,,~(.) 2-" II --,S=l ,yj II -
= n + l . We choose, again, sequences {n(j)}, {p(j)} and {q(j)} such that (2.2) and (2.3) are satisfied in addition to the following q(k)
(2.4)
b~(k)xj [1 >
~,
1 for k >_-1.
j =v(k)
Put n(1)= 1 , p(1)= 1 and choose q(1) so large, that I] v/'~jq o=)p ( 1 ) ~~,.(J)~ II > 1. j "~j [[ n(k + 1) and p(k + 1) are also chosen as in the first part, and q(k + 1) is so large that [[ --j ~'~q(k+l) K n ( k + l ) Xj [[ > 1. (This construction is possible since {y~(")}~°=1 =p(k+1)uj is a basis in [y~t,)]~o=~ and satisfies (a). By Lemma 2.1, if {g~}~°°=1 denotes the sequence of biorthogonal functionals of {yTtn)}°°=~,then []g[' []= 1. It follows that for all natural i and n Ibm'I = I gT(Eb~Y](")) [ < 2 - " < 1 .) By (a) and Lemma 2.2 q(k)
oo
z b "y;
j = p(k)
II <=II j z= 1
<=
= p ( o b,(O~ ~ "'J] It follows that ~z~ 1( vq(o ~ =p(Ob"(°,,~("(°)~ ~ yJ ~ converges while ]~=1( oo ~x'~(o certainly does not, by (2.4). But by (2.3) the sequence )~~y,(n(O)X j 5p(i) ~_ j ~_q(i),i>= 1 , forms a normalized Nock-basis; it follows that the last block-basis is not equivalent to the basis { ~}~= 1 a contradiction. This completes the proof of Lemma 2.5. 3. The main theorem. oo be a normalized basis in a Banach space X . Then THEOREM 3.1. Let {X /}/=1 {z}i=ix o~ is perfectly homogeneous if and only if it is equivalent to the unit-vectors basis of Co or of lp for some p > 1.
Proof. The " i f " part is obvious, since the unit-vectors bases in Co and lp are perfectly homogeneous. Let us prove the other part. By Lemma 2.4 {X z}zoo= 1 is an unconditional basis. By [3] p. 73 Theorem l(v) we may assume that {x/}/~l satisfies (a), hence, by Lemma 2.5 it satisfies the following property: (b) There exists a real M > 1 such that for every normalized block basis oO {Z ~}z=l, n >~--" 1 and real al,a2,'",an " aizz Define for k > 1
_"
~
]
1966]
PERFECTLY HOMOGENEOUS BASES IN BANACH SPACES
269
k
(3.1)
2k =
~ x~ t == 1
It follows from (a) that for k > 1 (3.2)
2k+i > 2,.
By (b), for every increasing sequence {p( i)}i"= 1 of positive integers
(3.3) It follows that n
ilk
Ilk- [
[=
(We substitute for z i in the right side inequality of (b) the normalized block [I ~i=1 .k-1 xy+(,_l)ilk- , []-1 . ( ~ yIlk-, = l xj+(,-i),k-~) and use (3.3).) Using 3.4 we can prove by induction that for every natural n and k (3.5)
M2k.
xi k >
xi .
i=l
i=
On the other hand, by the left-side inequality of (b) and by (3.3) n
ii k
~ l x'
< M2"
i-
ilk - 1
ZlXJ " i=
~1 X/ -1
.
j=
Again it follows by induction that for every natural n and k (3.6)
~ 1 x'
< M2k"
ii,,Yl x, "
(3.1), (3.5) and (3.6) yield (3.7)
M -2k" 2nk < 2~ < M 2k" 2ilk for every n and k.
For any natural N, n and k let h = h ( N , n, k) be the non-negative integer for which Nh<= nk < N h+l' By (3.2) and (3.7) h.log2~ < log(M 2h. 2~h) = 2 h " logM + 1og2Nh __--<2h" logM + log2ilk < < 2 h " logM + log(M 2k. 2~.) = 2h. logM + 2k. logM + k.log2,.
Since h < k" log n" (logN)-1 =< h + 1, we have ( k . l o g n . ( l o g N ) - 1 - 1).log2s < 2k.logn.logM(logN) - i + 2k. log M + + k-log2..
270
M. ZIPPIN
[December
Dividing by k log n and passing to the limit as k ~ oo we get (3.8) (log 2u)" (log N)- i < (2 log M)" ((log N) - 1 + (log n)- 1) + (log 2~). (log n)- t. By interchanging the rfles of n and N we get (3.9) (log 2n) (log n)- 1 < (2 log M)- ((log N ) - 1 + (log n)- 1) + (log 2u) (log N)- 1 . By (3.8) and (3.9) ](log 2~) (log n)- 1 _ (log 2N) (log N ) -
1 I =<
(2 log M) [(log n) - t + (log P 0 - 1 ]
therefore the sequence {(logX,)(logn)-l}~=l converges to a limit c, and since 1 < 2~ < n, we get that 0 <_ c <_ 1. Passing to the limits as N -+ oo in (3.8) and (3.9) we get: clog n < 21ogM + log2~ = log(M22,) and log(M-2"2n)=< c logn, hence, for every n (3.10)
M-2
nc ~
;Ln ~
M 2 . nc "
If c = 0 then {2.}.= oo x is a bounded sequence, therefore by Lemma 2.3 X is isomorphic to Co. If 1 __>c > O, put c = 1/p. We have
M-2"nl/P< il ~ xs li <=M2" n l/P"
(3.11)
i=1
Let rs be any positive rational number for I < i _< n and assume that r1 = m - x- ks, where m and ks are positive integers. It follows from (a), (b), (3.11) and (3.3) that n
o1 ) IIII 1--1 ,:,,x, =
=
S=l
S=I
> M -3 •
U
kl
j=l
~ xj+ X k. i=1
j=l
j=l
"
j+Z=t.
m=t
(We substitute in (b) for zs the normalized block
j=l
,.=t
\j=l
m-t
By (3.3), (3.12) yields (3.13)
]l 1~ r~'px' l] > M-*" m -x/'' il 1=1
n k s. By (3.11) where k = 2~=1
S= 1
xs
1966]
271
PERFECTLYHOMOGENEOUS BASES IN BANACH SPACES
[ ~ r~i'xil > M-6"(~ ki)l/P'm-l/'2M -6(~ )'/" I
--
=
--
= 1 ri
"
Hence, using (a), it is easily proved that for any real ai, a2, "", a.
(3.14)
II ;~ a'x'll ~
M-6"
(n)
~ la'l' 1,,.
i=1
t
1
Similar arguments yield the following
1
t=I
<_-M2 . m - l / p .
"xl
__<
i=1
<M3"m-'IP'II,:,~ [[l'j~_-' [1 ii" '- li-' (k' ~.7 '-'']
=
,x,
k
__<M,.,,,-,,,.
,,,
.
r. x , + x , .
j=l
Mo.,,-,,,.
m=l
(
"
1XJ+
[ .
,<,)" = Mo. iS,,,:
,ll/p
(The notations in (3.15) are the same as in (3.12).)Again, by (a), for any real al,al,"',an
t=1
i=1
(3.14) and (3.16) show that {xt}l~=l is equivalent to the unit-vector basis in This completes the proof of Theorem 3.1.
Ip.
REMARK. Using the deep result ofA. Dvoretzky [4] A. Petczyfiski and I. Singer proved in [6] the following PROI'OSmON 3.2. Let E be an infinite-dimensional Banach space with an unconditional basis in which all normalized unconditional basic sequences are equivalent. Then E is isomorphic to 12. Proposition 3.2 has the following alternative proof: By Theorem 3.1 E is isomorphic either to Co or to Ip for some p > 1. For 2 ~ p > 1 one can construct in lp a subspace isomorphic to the space (El 0) E2 ~)...)p, where Ek denotes the k-dimensional euclidean space. This can be done without using [4] (see e.g. [5].) The space Y = (El @ E2 0)"')p has an unconditional basis non-equivalent to the unit-vectors basis in Ip, l < p # 2 . In fact, if ts~'~"("+t) ""iS i = {r(n- 1 )n+ 1 plays the r61e of the unit vectors basis in E, c Y, n = 1, 2,..., the sequence {X i}ioo= 1 forms an unconditional normalized basis in Y. If it were equivalent to the unit
272
M. ZIPPIN
oo vectors basis {e ~}i=1 in lp we would get that, for some fixed M > 1, every natural n and any real a l , a 2 , . . . , a n n ae II Z,=I ,,
_-< I1 Z7=1 a,x,+~,n 1,o II~ M rl Z7~1 aiei [1' which is known to be false, since ~ ,n= l a i e ,ll = ~ r = l l o , lP) ~'' while II is equal to ( ~ 7 = 1 [a,l~) ~ It follows that there exist normalized unconditional basic sequences in Ip which are not equivalent to the unit vectors basis. Similar basic sequences can be easily constructed in c o and in It. It follows that E is isomorphic to 12. M-
1
~7=la,x,+~,~-.olJ
REFERENCES 1. C. Bessage and A. Petczyfiski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958); 151. 2. F. Bohnenblust, Axiomatic characterization of Lv-spaces, Duke Math. J. 6 (1940), 627. 3. M. M. Day, Normed Linear Spaces, Springer-Verlag, 1962. 4. A. Dvoretzky, A theorem on convex bodies and applications to Banach spaces, Nat. Acad. Sci. U.S.A. 45 (1959), 223. 5. A. Petczyfiski, Projections in certain Banach spaces, Studia Math. 19 (1960), 209. 6. A. Pelczyfiski and I. Singer, On non-equivalent bases and conditional bases in Banach spaces. Studia Math. 25 (1964--5), 5. THE HEBREW UNIVERSITY OF JERUSALEM