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PUJARA, Lakhpat Rai, 1938- SPACES AND DECOMPOSITIONS IN BANACH

SPACES.

The Ohio State University, Ph.D., 1971 Mathematics

University Microfilms, A XEROX Company, Ann Arbor, Michigan

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED Z SPACES AND DECOMPOSITIONS IN BANACH SPACES P

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Lakhpat Rai Pujara, B.A. (Hons.)* M.A., M.S.

******

The Ohio State University 1971

Approved By

) D a * r J p t Adviser Department of Mathematics ACKNOWLEDGMENTS

I would like to express my sincere thanks to ny adviser

Dr. David W. Dean for his guidance, help and constant encouragement.

I would also like to thank Dr. William J. Davis for his help and guidance. Finally I would like to thank my wife Nimmi and son

Tiriku who had to face a lot of difficult days during the course of my graduate work at Ohio State.

ii VITA

May 19, 1938...... Born- Hussein, India (now West Pakistan)

19^9...... B.A. (Hons.) University of Delhi, India

1961...... M.A. University of Delhi, India

19&L -1 9 6 5...... Lecturer in Math., Hans Raj College University of Delhi, India

1965 - 1967...... Teaching Assistant, Department of Mathematics, The Ohio State Univer sity, Columbus, Ohio

1967...... M.S., The Ohio State University, Coluabus, Ohio

1967 - 1968...... Teaching Assistant, Department of Mathematics, The Ohio State Univer sity, Columbus, Ohio

1 9 6 8 -1 9 6...... 9 Dissertation Fellow, The Graduate School, The Ohio State University, Columbus, Ohio

1969 - 1971...... Teaching Assistant, Department of Mathematics, The Ohio State Univer sity, Columbus, Ohio

FIELDS OF STUDY

Major Field: Mathematics

Studies in Algehra: Professor M.L. Madan

Studies in Analysis: Professors R. Bojanic, B. Baishanskl

Studies in Topology: Professor Norman Levine

Studies in Functional Analysis: Professors David Dean, W.J. Davis

iii TABLE OF CONTENTS

ACKNOWLEDGMENTS......

VITA ......

INTRODUCTION ......

CHAPTER

I. Notation, Definitions and Known Results . . . .

II. Some Results about £ spaces...... • ]P III. Some Finite Dimensional Subspace Properties . .

IV. Some Unsolved Problems......

BIBLIOGRAPHY ...... INTRODUCTION

spaces were introduced by Lindenstrauss and Pelczynski in

[13]. A Banach space X is called a ^ space if given any finite dimensional sub space BcX, there exists a finite dimensional subspace W such that BCKX and d(W, ^ n = dim(W). Jr The characterization and classification of a Banach space in terms of its finite dimensional subspaces is a technique that has proved very fruitful in recent years in solving problems in Banach spaces.

In [11], Lindenstrauss used this technique for the extension of operators and other related problems. In [1 5]» using this approach,

Michael and Pelczynski attacked some problems about c(k) spaces, and in [3 ], Davis, Dean and Singer employed this technique to study complemented sub spaces of a Banach space. In [1*0, Lindenstrauss and Rosenthal established a very deep result viz. the principle of local reflexivity which states that every finite dimensional sub- 4H(‘ space of X is 'close' to a finite dimensional subspace of X.

Recently in [7 ], Johnson, Rosenthal and Zippin developed this technique in a very deep way with the work culminating in solving affirmatively the outstanding problem that if X has a basis, then

X has a basis.

In [1 3 ], the authors proved the main representation theorem about spaces viz. every space is isomorphic to a complemented subspace of some L^(ti) space, 1 < p < «» and for p = 1, if X is complemented in X . In this paper, they also studied p-absolutely summing operators originally introduced by Grothendleck for p = 1.

Combining the finite dimensional subspace approach and some powerful results about p-absolutely summing operators, they proved that every bounded unconditional basis of a ^ space space) is equivalent to the usual basis of A^(co).

In 1969, Lindenstrauss and Rosenthal [lU] solved some of the problems left open in [133. For example, they proved that every complemented subspace of an L (n) space is either a Hilbert space P or a space, 1 < p < » and for p = 1 , «», it is always a £^ space.

This result together with the main representation theorem about £ spaces implies that if X is a £ space, then X is a £^ space,

— + ^ =1. Recently in [ 7 ], the authors have proved that every separable £ space has a basis. P There are still many open problems in the theory of spaces.

For example, the number of different (upto isomorphism) infinite dimensional separable £ spaces is unknown 1 < p < *». Only six different such £ spaces are known [1 8]. Bessage and Pelczynski P (for p = “ ) and Lindenstrauss (for p = l) showed that there are infinitely many different separable infinite dimensional £ spaces. Jr We shall discuss some more unsolved problems in Chapter IV.

In this paper, spaces are studied. We also study some properties that one might naturally ask in terms of finite dimen sional subspaces of a Banach space to study the structure of that space. In Chapter I, the notations and definitions are given and

some known theorems are stated which are used in the subsequent

chapters. Also a new proof of a theorem of Lindenstrauss and

Pelczynski [1 3 ] is given which says that every bounded unconditional

basis of an Zm space is equivalent to the usual basis of eQ.

In Chapter II, £ spaces are studied. Proposition 2.1 estab- J r lishes a property of Z^ spaces that is generally known but which

nowhere has a very explicit statement in the literature. The

technique of proof is frequently used in this work. Theorem 2.h

gives a sufficient condition for a subspace of A■ to be complemented

in in terms of the 'support' of a basis of the subspace. Theorem

2.6 is the main result of this chapter which says that if a Banach

space X is non-£ , then it has a sub space Y which is non-£ and P P which has a Schauder decomposition into finite dimensional subspaces.

This result sheds more light on Proposition 7.2 [13]. Lemma 2.10

supporting Theorem 2.6 is of independent interest. Theorem 2.11

gives a sufficient condition for a subspace of A^ to be a Z ^ space.

In Chapter III, we introduce two new properties in Banach spaces.

A Banach space X has the X-finite dimensional subspace basis property

(X-f. d. s. h) if for any finite dimensional subspace B c X, there

exists a finite dimensional subspace W such that B c w C X and W

has a basis with basis constant less than or equal to X. The intro

duction of this property is further Justified by the fact that every

Banach space with a basis has f. d. s. b. .

Using the principle of local reflexivity, it is proved in ftft Proposition 3 .6 that if X has X- f. d. s. b.* then X has

(X + e) - f. d. s. b. for every e > 0. As a corollary to Theorem * 3.18* we obtain that if X has a basis* then X has f. d. s. b. .

Theorem 3*12 states that if a Banach space X does not have f. d. s. b. then it has a subspace Y which does not have f. d. s. b. and which has a Schauder decomposition into finite dimensional subspaces. In this chapter, we also study spaces [Definition 3.15]* The study of this property is motivated by the f. d. s. b. property and the

property. Every space with a basis is a space. Using two lemmas of [7 ]* it is proved that if X is a separable space* then

X is a B space for some n [Theorem 3*18] and if X is a B. space, [i A. ft ft X has n metric approximation property* then X is a Bf space for some r [Theorem 3-21].

Finally in Chapter IV* we discuss unsolved problems which have arisen in the course of this study and whose solution would shed great light on the structure of Banach spaces in terms of finite dimensional subspaces. CHAPTER I

NOTATION , DEFINITIONS AMD KNOWN RESULTS

In this paper, we deal with real Banach spaces. If X is a

Banach space, then U(x) denotes the unit hall of X. By a subspace of X, we ali/ays mean a closed linear subspace. By an operator, we understand a bounded linear operator. We denote the dual of X by

X and, as usual, identify X with a subspace of X .

Definition 1.1

By a projection P on a Banach space X, we mean an operator p on X such that P = P. A projection with a finite dimensional range is called a finite rank projection.

Definition 1.2

A subspace Y of a Banach space X is said to be complemented in X if there exists a projection P from X onto Y.

Notation 1.3

If (x,)? - ((x.)^ ,) is a sequence of elements in a Banach 1 1 =lL 1 I n i space X, the subspace of X spanned by ?=c) wl u be denoted by will be written as [x^ if no confusion arises.

Definition 1.^

Tito Banach spaces X and Y are said to be isomorphic if there 5 exists an invertible operator from X onto Y, then we write X ~ Y.

Definition 1.5

If Y is a subspace of a Banach space Xj then X/Y denotes the quotient space* Y has finite codimension in X if X/Y is finite dimensional.

Definition 1.6

A Banach space X is called an injective space if there exists a real number X such that for any Banach space Z, Z^X, there is a projection P from Z onto X with ||p|j < X.

Definition 1.7

A sequence (x^) of a Banach space X is called a basis for X if for each x € X, there exists a unique sequence (a.^) of real n numbers such that x = £ a^x^ = lim £ aixi* ^ se(iuence (x^) in. n -» » i =0. a Banach space X is called basic if it is a basis for the subspace that it spans in X. A sequence (x^) is called an unconditional basis for X if x = £ a^x^ converges unconditionally to x.

Theorem 1.8

A sequence (x^) in a Banach space X is a basic sequence if and only if there exists a real number k such that

H £ a x || < k || £ a x || (l) i^L i=l 1 1 for all sequences (ai), r < s. The infimum of k appearing in (l) is called the basis constant for (x^). Definition 1*9

If (x^) is a basic sequence in a Banach space X, then the sequence (f^)in [x^] 3 f^(Xj) = 5.^, i 7* j, i,j = l,2,3j«»» is called biorthogonal functionals sequence. It is well known and is, in fact, a simple consequence of Theorem 1.8 that they constitute a basic sequence.

Definition 1.10

Let X be a Banach space. A family (Bn )^_Q_ subspaces of

X is called aScliauder - ■ ■ - decomposition -■ ■ ■ for X, written as X - Z © X_ n if every x € X can be uniquely written as x = Z xn, xft € Xn» It is well known that the projections P such that P (Z x ) = x are m m n m uniformly bounded.

Definition 1.11

Let (Bn ) be a sequence of Banach spaces. We define for p > 1,

B = (Z © B , n = £(b ) : b € B and Z lib |IP < • n7J?p(resp. cQ) v n' n n 11 nH (resp. ||bn|| -> 0 )}.

In B, we define 1 ll(bn )|| = (2 l|bnl|P)P (resp. max ||bj|), n then B is a Banach space [16].

Notation 1.18

By L (p) = L (Cl, o, p), 1 < p < «>, we denote the Banach space P P of equivalence classes of measurable functions on the measure space

(fl, o, p) whose pth power is integrable if 1 < p < 00 and essentially bounded if p = ». If (Q, a, u) is the usual measure space on [0,1] we denote L (n) by L . If 0 is discrete and h (Cy }) = 1 Vy € n, P P

\re denote I. (n) be I (Cl) and if Cl is countable, then by A * If P P P n = Cl, 2 , ...,n} with counting measure, then L (n) is denoted by P (n) ^ P ■ The subspace of J&w spanned by sequences which vanish at ** is denoted by c , o

Definition 1.13

Let X and Y be Banach spaces. Define

inf(]|T|| Hi"1 !!): T = X -* Y is an (T isomorphism onto

« if T is not isomorphic to Y.

Also X and Y are said to be e-isometric if there exists an onto isomorphism T: X -* Y such that

(1 - e ) M < ||Tx !| < (1 + e ) M , x € X.

If e = 0, then T is an isometry and X and Y are said tobe

isometric. We state, without proof, the following useful relation

connecting e-isometry with d(X, Y).

Lemma l.jif

d(X, Y) < X if and onlyif X and Y are isometric.

We shall use this lemma in computations throughout this paper.

As was mentioned in the introduction, spaces were introduced by Lindenstrauss and Pelczynski in [1 3] in 1968 and in [l4] Lindenstrauss and Rosenthal solved some questions left open in

[13]. We state (without proof) below the main theorems of [1 3 ] and

[1^] which are relevant to the contents of this paper.

Definition 1.15

A Banach space X is called a £ . space, 1 < p < », if given Pi * — any finite dimensional subspace B c X, there exists a finite dimen sional subspace W s.t. B C \j C X and d(w, ^ where n = dim(w).

X is called a £ space if it is £ . for seme X. p P jX

Examples l.l6

(1) Every is a £p space. The verification of this fact is non trivial. We would discuss this result in Proposition 2.1. Here 1

(2 ) © i , 1 < p < is a £ space [1 3 ]. ^ P P

(3) (i2 © ig © ... )jj is a * space 1 < p < « [1 3 ]. «Er *

(4) In [l8], Rosenthal has given examples of £ spaces (p > 2) P spanned by certain sequences of random variables on [0, l] which are isomorphically different from the above examples.

Theorem 1.17 [13]

(a) Let X be a £ space, 1 < p < w. Then X is isomorphic to a P complemented subspace of some Lp(ii). If X is separable, then it is isomorphic to a complemented subspace of L^.

(b) For p = 1, °°, X is isomorphic to a subspace of some .

(c) For p = 1, », X is isomorphic to a complemented subspace of ■JHfr some X(p(|-i) if X is complemented in X . 1 0 note 1 .3.8

For p = 1 , it is not always true that if X is £ , then X P is isomorphic to a complemented subspace of some L^(u). In fact, cq is an space and if cq is complemented is some L^u), then cq woult^ "be injective as every Lw (n) space is injective. But this is known to be false as no separable space can be an injective space.

For p = 1, Lindenstrauss [9] bas given an example of a space which is a subspace of and which is not isomorphic to a comple mented subspace of any L^(^).

Theorem 1.19 [8]

Every infinite dimensional £ space, 1 < p < «, has a comple- P mented subspace isomorphic to Jfc •

Note 1.20

Theorem 1.19 is not true for p = “. In fact, c([0,l]) is a

space but it has no subspace isomorphic to

Now we state some theorems connecting the structure of X and * X if X is a £ space. Jtr

Theorem 1.21 [l4]

(a) Every complemented subspace of a I» (n) space is either a

Hilbert space or a space, 1 < p < ».

(b) For p = 1, every complemented subspace of a Lp(n) is a

£p space.

Remark

For 1 < p < “, every L (w) has a complemented subspace isomorphic P to a Hilbert space. In fact, the space spanned by Rademacher functions 11 in 1^ is a complemented subspace of [l6]. Also using Theorem

1.17» Theorem 1.21 and the principle of local reflexivity, it follows that Theorem 1.21 is true if we replace L (p) by any £ space. ir Jr As a simple corollary to Theorem 1.17 > Theorem 1.21 and the principle of local reflexivity, it was obtained that

Theorem 1.22 [l*f] * X is a £ space if and only if X is a Z space, 1 < p < m j P P i . l - x . p 1 The question of "nice" finite dimensional superspaces of finite dimensional subspaces in a £ space was also settled in P [14] viz.

Theorem 1.23 [l4]

Let X be a £ space, 1 < p < w. Then there exists a constant P X such that for any finite dimensional subspace B of X, there exists a finite dimensional sub space W 3 B C W C X, d(W, < X, n = dim(w) and there exists a projection P: X -* W, ||pi| < X.

One must mention the following:

Theorem 1.24 [7 ]

Every separable £ space has a basis, 1 < p < <°. 5

We close this chapter Trith a new proof of the following theorem of Lindenstrauss and Pelczynski [133.

Theorem 1.23

Let (x^) be a bounded unconditional basis of an £tt space X. 12

Then (x^) is equivalent to the usual basis of c q»

We need the following lemmas:

Lemma 1.26 [12]

Let (y^) te a bounded unconditional basis of a space.

Then (y\ ) is equivalent to the usual basis of A^.

In [13 ], the authors proved this lemma using some strong

results about p-absolutely summing operators.

Before stating the next lemma, we fix a notation and give

a definition. If (x^) is a basic sequence and (f^) are the

corresponding biorthogonal functionals, we shall write (x^; f^).

Also a basic sequence (x^) is cal led bounded if there exists positive numbers m and M such that

m < fix. II < M i = 1,2,.. •

Lemma 1.27

Let (x.; f .) and (y.; g.) be basic sequences. If (x.) is

equivalent to (y^), then (f^) is equivalent to (g^).

PROOF:

The lemma is known. We include its proof.

First of all, ire say (x^) is equivalent to (y^) if T: [x^ ->

Cy±3 3 ^ C x ^ = y± is an onto isomorphism.

Since (x.^) is equivalent to (y^), there exists T: [x^] -> [y^] 5 * * # Tfx^) = yi and T is an onto isomorphism. Then T : [y^ -» [x^] is

such that

(1) 13 Also ^(Xj) a 6j, j (2)

Since (xj) is a basis for [xj], from (l) and (2), we conclude & that T (g^) = f^. Since T is an isomorphism*

T*: [%] - Cq]

is an onto isomorphism. Hence the lemma.

Lemma 1.28 (James)

Let (x±) be an unconditional basis for a Banach space X. If

(x^) is not shr inking* then X has a complemented sub space isomorphic to 4^.

Proof of Theorem 1.25

Assert that ( x ^ is shrinking. If (xi) is not shrinking, then by Lemma 1.28, X has a complemented subspace Y isormorphic to 4,. By Theorem 1.21, being complemented in X, Y is a £ space. Thus we get a contradiction implying that (x^) is shrinking.

Therefore, (f^), the functionals biorthogonal to (xi ), constitute a bounded unconditional basis for X* [19] &nd X* is a space

[Theorem 1.22]. Therefore, by Theorem 1.26, (f^) Is equivalent to the usual basis (h^) of 4^, Let (g^), (k^) denote the functionals biorthogonal to (f^) and (h^) respectively. Therefore, by Lemma

1.27, (gt) is equivalent to (k^. But ( g ^ is equivalent to ( x ^ and (k^) is equivalent to (e^, the usual basis for cQ. Therefore,

(xi) is equivalent to (e^). Q. E. D. CHAPTER II

SOME RESULTS ABOUT £ SPACES ______P______

In this chapter, some results about 5^ spaces are proved.

The main result of this chapter is that if X is not a £ space, Jfcr then there exists a sub space Y of X which is not a £p space and which has a Schauder decomposition into finite dimensional subspaces.

Proposition 2.1

Let X be a Banach space such that X = U Bn where (B n ) is n an increasing sequence of finite dimensional subspaces of X and there exists a constant X > 1 such that

(a ) d(V *p ' ^ X > mn = then X is a ^ space. Conversely if X is a separable £ p ^ space, then X = O X for some B 's as above, n n n

PROOF:

Let B be a finite dimensional subspace of X and let B = where x^>x2,... ,xq are linearly independent. Let e > 0 be arbitrary.

There exists an integer N and (y^)^^ c: such that

ll^i - y iH < e, 1 < i < n. (l)

14 15

Also there exists 6 > 0 such that

|| £ a x H > 6 £ |a | (2) i=l 1 1 i=CL 1 for all finite sequences (a. )? - of Scalars. We split the proof 1 x of the theorem into several steps.

Step I

Assert that is a linearly independent set for e small enough. n n Consider || £ a.x. - £ a.y || i=SL i=l 1 n < Z |a. | ||x. - y.|| i=l 1 1 n < « £ |a. | by (l) i=l

< | || £ a^x^|| by (2) (3*) i =&

Thus

II 2 a y || > (1 - -jJH 2 |1 (3) i=l 1 1 0 i=i 1 1

Since e is arbitrary, let e < 6.

Then n n 2 a y = 0 =» £ ax . = 0 by (3 ) i=l i=l

^ s 0^ 1 ^ i ^ n as are linearly independent =* y^ * . . *yn are linearly independent* 16

Step II

Let Bjj - S E for same E* Asser't: 'fchat E ^ ^xi^i~L = £°}* n Svgppose there exists x = Sax., ||xl| = 1 and x € E. Let i=i 1 1

P: BN ** ^ 1 ^1 =3. Projection such that P(E) = (O). By hypothesis, n n E a.x - E a.y. (: B„. Therefore, i=d i=l 1 1 a

« v p>tivi - ivi>n

= || E a x || = 1 (4 ) i=l 1

Also

||(I - P)( E a x. - E a y.)|| i=d i=»l X

< ||l - P|| by (3*), (5)

Thus from (U) and (5)j we get 1 < H1*" PU *f 811(1 this is false

choosing e small enough. Thus E H ^ and 80 w = E ®^xi^i=4 is well defined.

Step III

Assert that d(W, B^) < 1 + ^ for every 1] > 0. Indeed define

T: W -* Bjj as follows: 17 n n T( Z a.x. + e) = Z a.y. + e i =1 i=a where n Z a.x, € [x. ]n and e € E. 1=1 1 1 i^L

Then n n ||T( E + e) - ( ^ a ^ + e)|| i=d i=a

- i! 2 a i(x i “y^)!! i=a 1 1 1

< E K |||x, - y || 1=1 1 1 1

n S 6 !l 2 aixill (l) and (2)* 1=1

< f H 2 *1*4 + e)|| 1=1 where Q: W -» Is a projection with Q(E) = {O}.

< | INI II s + e|| i =1

Thus _ fi , . 1 + ! w d(W, B j < ----\---- . “ 1 - -f llQ ll

Finally choose e such that, in addition to satisfying the conditions of Step I and Step II, 18 i * | IN I < i + ? . i - f INI

(m ) (m ) Then d(W, Jdp n ) < d(W, V d(BN> j&p n )

< (l + ^-J)X = X + T|.

Thus X is a £ . - space for every 1] > 0. P jA,+ iI

Conversely let ( x ^ he a dense set in X. Since X is £p there exists a finite dimensional subspace such that x^ € c x ( n O and d(B., J& ) < X. Again there exists a finite dimensional 1 p - m subspace Bg such that Xg] c Bg c x and d(Bg, £p ) < X. In general there exists a finite dimensional subspace Bn such that (m \ [B ,, x ] c b c x and d(B , I n') < X. Thus from the above n-1 n n n ^ construction, it is clear that there exists a sequence (Bn ) of finite dimensional subspaces such that x = C T V Bn C V l ' n ...... * n (mn) and d(Bn, £p ) < X. Q.E. D.

We obtain the following corollaries:

Corollary 2.2

Let (xi) be a basis for a Banach space X. Suppose for every n, [x., is contained in a finite dimensional subspace B of X i i-l n (%) such that d(B , & ) < X where X is a constant, then X is a £ n' p - p space. PROOF: lc Let B be any finite dimensional subspace of X, ^e a basis of B and e > 0. Then there exists an integer H such that

N II®i - £ a^XjH < e, 1 < i < k

N for some Kow the proof is completed exactly as in

Proposition 2.1.

The proof of the following result was suggested by and is an expansion of a remark in [lU, page 326],

Corollary 2.3

L^(n) is a Xp space for every e > 0 , 1 < p < ».

PROOF:

Let x1 #Xg,...,xn be a basis of an arbitrary finite dimensional subspace of L (n). Since simple functions are dense in L (p.), for P P T| > 0 , there exist simple functions y ^ y g , ... ,yn such that

llx^ " y ± \\ < i = 1,2,...,n.

As in the proof of Proposition 2.1, we can show that (y^)^^ are linearly independent. Let mi y = L 1 j=l J ij

“i where (A. .). ., l

mi n of the sets ((A. .). .)” .. Let D denote the subspace of L (p.) ij j =x X p spanned by the characteristic functions of the sets in A. Then (r) D is isometric to & where r = dim(D). P

Write D = Cy^3^^ ® ®

W = © E.

As in thvi proof of the proposition 2.1, we can show that W is well defined and d(W, Al J) < 1 + e for every e > 0 by choosing 11 Jr suitably. Q. E. D. CO Let 1 < p < « and (x.) be a sequence in A . Let x. = £ a. .e. —* x P X j ^ Xj J where (e.) denotes the usual basis of A . Define N, « (j: a. . ^ 0}. J P i ij

is called the support of x^. Pelczynski [l6] proved that if

Ki ^ = 1 ^ = then [x^] is complemented in

A . Using his result, we prove the following: P

Theorem 2,h

Let 1 < p < “ and (x.) be a sequence in A such that

K ~ U (N,ON,) is a finite set. Then [x. ] is complemented in A , i 4 * J ^ ‘P i #

PROOF:

Let y = £ a .e. where a . is zero if d does not appear d©c ^ ^ “•* in the support of x but belongs to K. Let z„ = x - y„ = E a .e. n “ nn^nj^nod definition of li^ that N* P nJ = f> if i t ,}. Thus by [16], [zn ] is complemented in S> . Let [y ] - B'. B* is a finite dimensional P » subspace as B* c ^ = B> say.

Assert that [x^] + B = [z^] © B. (l)

Clearly [z ] © B is well defined. Since z = x - y € [x l + B, n n n n n we conclude that [zn ] 9 B c fxn ] + B. (2)

Also x € [zj © B for all n, we conclude that [xn] + B C [z ] + B (3) n n n n

From (2) and (3)> we obtain (l). Since [z^] is complemented in t ^ 9

[z ] © B is complemented in A . Since [x 1 has finite codimension n p n in [xn ] + B, we conclude that [xq ] is complemented in Q.E.D.

Following [7], we give the following definition:

Definition 2.5

A Banach space X has the finite dimensional decomposition property (written as f.d.d.) if it has a Schauder decomposition

into finite dimensional subspaces. Now we prove the main theorem

of this chapter. 22

Theorem 2.6

Let 1 < p < °°. If a Banach space X is non-£p space, then it has a sub space Y which is non-i^ and which has f.d.d. .

The following lemmas are needed frequently below and in

Chapter III, The first and the third are known, [4], [21] and the second is generally known but is not anywhere explicitlystated.

We include the proofs here. The fourth lemma is of independent interest.

Lemma 2.f [If]

Let B be a finite dimensional subspace of a Banach space Z.

Then, for e > 0, there exists a subspace W of Z with the following properties:

(1) W Pi B = [o] and the natural projection of W ® B ■» Bhas norm < 1 + e.

(2) W has finite codimension in Z.

Proof of Lemma 2.7

Since B is finite dimensional, U(B ) is compact. Let 6 > 0 be arbitrary. Let (g^)^b e a 6-net on U(B ) and let f^ denote the Hahn-Banach extension of g^ to Z, 1 < i < r. For each b € B,

define jbj = sup |f.(b)|. Assert |*| defines a new noun on B 1< i< r 1

equivalent to the given norm ||*|U Clearly | b | < H b | | (l1)*

ft Since B is finite dimensional, there exists gQ € U(B ) such that 23

go(b) = ||b||, l!e0llB =* i.

There exists some 1 < i < r, such that ||gQ - SjJIg < 6.

Therefore 3

||b|| = gQ(b) < |gQ(b) - gi(b)| + Ig^Cb)!

< II gQ - %IIB1W| + lb |

< 6 llb|| + lb] i.e. (1 - 5)]jb|| < |b| (2*)

From (l') and (2*)* we get

(1 - 6)l|b|l < |b| < M (3') which shows that | • | is a norm on B equivalent to the given norm.

r , Let W a (If. (0). Clearly \J has finite codimension in Z so i=a.1 that (2) is satisfied. Suppose there exists b € B Pi W, b ^ 0.

=> f^(b) =0, l < i we have

» ,i Ibi llbll 5 jTs - ■ x ~ i for 80,18 1 ___

IfjCb + w)| " — 4 ^ 5 ------> w € w

< jrj- llb + '*11

< 1^6 llb + “I! 2k

< (1 + e)||b + wll if

Thus the natural projection of W © B -* B has norm < 1 + e. Q.E.D.

Lemma 2.8

Let \J he a subspace of a Banach space Z such that W has finite codimension in Z. If W is a Z space, so is Z. P PROOF:

Suppose W has codimension n in Z. Therefore there exists linearly independent vectors x1,... ,x in Z such that Z = W © [x. j . x n i=l

Since any two finite dimensional sub spaces of the same dimension are isomorphic, 3 X > 1 9 d( [x, ]? _, < X. Direct sum of tiro Z "■ 1 1 ni P — P spaces being a Z ^ space [13 L we conclude that Z = W © [x^]^_^ is a

£ space. P

Lemma 2.9 [21]

Let E^, Eg be n dimensional subspaces of an (n+l) dimensional

Banach space 6, then d(E^, Eg) < 9*

PROOF:

By hypothesis, E^ H Eg is a subspace of (n-l) dimensions.

Therefore, there exists a projection P^: E^ *■» E^ 0 Eg such that

III - PjJI « 1, i = 1 ,2 . We can write E^ » (E^ f! Eg) © [x^] where 25

Pi(xi ) = 0, ||Xi || = 1, i = 1,2. Define T: -> Eg as follows:

T(e + coc^) = e + aXg, e € 0 Eg.

Then

||T(e + cu^Jll < ||e|| + |a|||x^||

= l|e|| + llaxjl

= Hp^e + 0^)11 + ||(X - P ^ fe + 03^)11

< (2 + l)||e + axjl

i.e. |]t J| < 3« Similarly ]|t "^J| < 3* Hence the lemma.

The following lemma says that if Y has finite codimension in Z,

and finite dimensional subspaces of Y have ’good1 finite dimensional

superspaces in Z, then finite dimensional subspaces in Z have 'good'

finite dimensional superspaces in Z. The technique of the proof is

used in Chapter III also.

Lemma 2 . 1 0

Let Y be a finite codimensional subspace of a Banach space Z.

Suppose there exists a constant X > 1 such that for every finite

dimensional subspace B c Y, there exists a finite dimensional sub

space W such that BCKZ and d(U, < X, then Z is a space.

PROOF:

He shall give the proof by induction on n, the codimension of

Y in Z.

Let Z = Y © [xQ] for some xq € Z and P: Z -> Y be the corresponding 2 6

natural projection with f|f {| < 2. Let E C Z tie any finite dimensional

subspace. If E c y, then it has already a ’nice* super space in Zs

If E Y, there exists a finite dimensional subspace E 1 c y such that

E c E' © [x q ]. By hypothesis of the lemma for n = 1, there exists a

finite dimensional subspace F, E 1 c F c z and d(F# < \. If P ™ xc € F, then E c E* S [xQ] c: p c z and we have a 'nice’ superspace

of E. Suppose xQ f- F, Since d(F, 4 ^ ) < there exists an

operator T: F *■» such that ||t|| < 1, ||t”*|| < 1. Define

S: F © [xQ ] -> as follows:

S(f + axo) = T(f) + oem+1 where (e . denotes the usual basis of Then ' i i d P

||s(f + ouc0)|| < IMI||f|| + l|aem+1||

< Ikll - K l l

= ||CC - $ (f + «*xo)|| + l|R(f + cxxo)||

where R: F © [x ] ** [x q ] is a projection with R(F) = (o} and ||r || < 1

< (2 + l ) |l f + oocjl =*||S|| < 3.

Also . m +1 IIS ( S a ^ ) ! !

||T_1( “ o^ ) + am+1x0|| Thus d(j? © [x ], < 3(1 + X). Thus the theorem is Jtr proved for n = 1. Suppose the theorem is known for n-1.

Finally let Z = Y O **et E c Y be any finite dimensional subspace. B y hypothesis of the lemma, for n, there exists a finite dimensional subspace W such that E c \j c z and d(W, < X.

VI 1 If W c Y 0 [x. we proceed to the next step, otherwise let 1 1~X

W ’ = [W, x ]. If W = W , we proceed to the next step. If If r W , n then ire can show, as in the first part of the proof, that

d(W, / m+1>) < 3 (1 + X). Jr

Let S: W -> be such that [|s|| < 1, IJs"1 !! < 3 (1 + X).

Write W* » G © txn J for some G C y $ 111611 A ~ s/G:G->^ m+1^ is an isomorphism of G into and its range L is an m dimensional subspace of Thus d(L, G) < 3(1 + X). By Lemma 2.9, d(L,/m ^) P . . P < 9. Therefore, d(G, < 27(1 + X). Thus E c G c Y © Cxi] J ^ .

The conditions of the induction hypothesis for n-1 are satisfied and thus Y © foj^iKL is a sPace* The fact that every one dimensional 28 space is a £ space and the direct sum of tiro £ spaces is a £ P P P space implies that Z = (Y © © [xn 3 is a space. This completes the proof of the lemma.

Proof of Theorem 2.6

Let (X ) he any increasing sequence of reals tending to infinity.

Since X is not a £ space, given X_, there exists a finite dimensional P subspace V7, of X such that for any finite dimensional subspace B_, 1 r \ A c B1 c x, d(B^, £ ) > X^. Choose one such B^. By lemma 2.7, given e > 0, there exists a subspace Y^ such that

(1) Y^ has finite codimension in X.

(2 ) y1 D b 1 = {0 ].

(3 ) the natural projection of Y^ © -» has norm < 1 + e.

By Lemma 2.8, Y1 is not a £ space. Since Y^ © B^ has finite codimension in X and X is not a space, Y^ © is also not

£ • Therefore, by Lemma 2.10, given Xg, there exists a finite P dimensional sub space Wg C y , such that for any finite dimensional

(®J subspace Bg, Wg c Bg c © B^, d(Bg, ^ ) > Xg. Choose one such

Bg c Y^. Suppose B^, Bg,...,Bn and * * * ,Yn-l have been c^osen such that

(i) dim(B^) < «, 1 < i < n

(ii) Y± c Y±-1, Yq = X , 1 < i < n-1

(iii)Bi+1c y a, 1 < i

(iv) The natural projection of (B^ © Bg© . . . © \) ®

B^ © ... © B^ has norm < 1 + e, 1 < n-1,

0 0 (v) d(Bi, Ap 1 ) > \± , 1 < i < n.

(vi) Any finite dimensional super space of in B ©Bg© ... ©

also satisfies (v), 1 < i < n.

By Lemma 2.7, there exists a subspace Y^ such that

(a) Y c Y i and has finite codimension in Y ,. n n- 1 n-1

(b) Yn O [B^Bg,...^] = CO}.

has norm < 1 + e.

Since Yn-1 is not a £p space, by Lemma 2.8, Y ^ is not a £p space.

Therefore, given Xn+^, by Lemma 2.10, there exists a finite dimen sional sub space Wn+1 of Yn such that for any finite dimensional (m _) subspac. Bn+1, Wn+1 c Bn+1 = Yn ® ^ ® B2 ® ... ® 1 ) >

\i+l*

We choose one such B contained in Y . This, by induction, n+1 n completes the choice of the sequence (Bn ) and (Y^). Let Y = 2 ® ®n *

By standard arguments, we can show that Y is well defined and is a closed subspace of X. Also the way we have constructed B^'s, it is clear that if D is a finite dimensional subspace such that n ' n / B C D c Y, then d(D , & " ) > X where s = dim(D ). Since XX IX XX p IX XX XX

\ -» », we conclude that Y can’t be a £ space. Q.E. D. n p

Remark

Ercm the proof of the above theorem, it follows that if every separable sub space Y of a Banach space X can be contained in a separable sub space Z which is a space, then X is a ^ space.

This is converse to [Proposition 7*2 [13]]*

Remark

It is not k n o v m that if a Banach space has f.d.d., then it has a basis [3 ]. It is worth noting that in the proof of Theorem 2.6, we had a lot of freedom in the choice of B 's. In fact, any n finite dimensional superspace B^ of Wn would have been good enough for our purposes. Thus if the space X (as in Theorem 2.6) has the property that every finite dimensional subspace E of X can be con tained in a finite dimensional subspace B of X such that B has a basis with basis constant less than or equal to (i for some constant

|4 , then we could construct a subspace Y of X such that Y has a basis and Y is non-£p. This above mentioned property is of independent interest in general Banach spaces. We shall study this property and other related properties in detail in Chapter IV.

In the next theorem, we obtain a sufficient condition for a subspace of ^ to be a space. 31

Theorem 2 . 1 1

Let X be an Infinite dimensional subspace of X^, 1 < p < «, or cq. Suppose there exist real numbers X^, Xg such that for any finite dimensional subspace B of X, there exists a finite dimensional subspace E of Jl (respectively cQ) with the following properties: i) there exists a natural projection P^: E ® B © A -» B with norm Xi < » 1 ii) d(E, < \s , dim (E) = n. iJ Cm Then X is a space. P PROOF:

By Lemma 2.7, there exists a subspace Y c X 3 Y has finite codimension in X, Y D B = (o) and the natural projection of Y © B -* B has norm < 2. Since every infinite dimensional subspace of X has ir a sub space e-isometric to X^, 1 < p < 00 [16], we see that 3 Z c Y 3 d(Z, A ) < 2. Thus 3 z c x such that Zf!B a (o) and the natural projection Q: Z © B -* B has norm < 2 and d(Z, X ) < 2 . By hypothesis, there exists a finite dimensional subspace EofX3E=B©A and the natural projection P: B © A -» B has norm < X^, and d(E, X^ ) < Xg , n^ = dira(E). Let A' be a finite dimensional subspace of Z 5 d(A',A)

< 2. This is possible as d(Z, X^) < 2, Let R: A' -* A be the isomorphism such that ||r|| < 1, Hr**1 ]! < 2. Let W = B © A1. Clearly

W is well defined. Define T: W ■> E as follows:

T(b + a') = b + R(af), b € B, a' € A*.

Then 32

||T(b + a* )|j < ||b|| + ||R(a')||

< M l + ||a*||

= lfc(b + a*)II + 11(1 - Q)(b + a')ll

< ( N 1 + III - Q||)||b + a* 1|

< 5||b + a ?||

- M l < 5 Also

+ a ) || = ||b + R"1 (a)H

< M l +

= ||p(b + a)|| + ||(I - P)(b + a)||

< ( 1 + ZljjW* + a )

- I K 1 !! < (1 + 2 xx )

{*.) Hence d(W, £ x ) < 5 ( 1 + 2XX )X2

The proof in case of cQ is the same.

Remark

The theorem would still he true if it was hypothesized that for any finite dimensional subspace B of X, 3 a finite dimensional sub space c X 9 B c c X and c e c satisfying (i), (ii) in the theorem. 33

Proposition 2 . 1 2

Let X “be a £ space, 1 < p < » and Y be a Banach space. Then

X c y is complemented in Y if and only if there exists a constant

X such that given any finite dimensional subspace B of X, there exists a finite dimensional subspace 3 B c tf c X, d(W, 4^”^) < *■ and there exists a projection F: Y -» W, ||pj| < X.

We need the following result.

Lemma 2.13 [10]

Let Z be a reflexive Banach space and ZCY, a Banach space.

Then Z is complemented in Y if there exists a constant X such that given any finite dimensional subspace B of Z, 3 a finite dimensional subspace W 3 B c u C Z and there exists a projection P: Y •* W,

M <

Proof of Proposition 2 . 1 2

Let X be complemented in Y and let Q: Y ■» X be a projection.

By definition of and Theorem 1.23, 3 a constant \i such that given a finite dimensional subspace B of X, there exists a finite dimensional subspace W, B c H c X, d(w, 4 ^ ) < H and P: X -» W is a projection# lJp|! < p. Then PQ: Y -* W is a projection and ||pq!| < M-llo)]*

The converse is an immediate consequence of Lemma 2.13. Note that for this part the assumption that X is <2^ is superfluous. Q.E.D.

Theorem 2 . 1 4

Let X be a space with a bounded unconditional basis (x^) such that no sub space of X is isomorphic to J&g, 1 < P < “ j P Then X has a Schauder decomposition 2 © Xn where each Xn is isomorphic to

We need the following lemma:

Lemma 2 . 1 5

Let (x^) be a bounded unconditional basic sequence in L^ such that [ x ^ is complemented in L^, 1 < p < "• Then (x^) has a sub sequence which is equivalent to the usual basis of J&2 or

Proof of Lemma 2.15

For p > 2, the lemma is known by a result of Kadec and Pelczynski

[8]. Now consider 1 < p < 2. Since [ x ^ is complemented in L^, the functionals (f^) biorthogonal to (x^) can be, in a natural way, extended over L^ in such a way that [f^] is isomorphic to a comple mented subspace of L , — + — = 1. Since q > 2, by [8], (f.) has P a subsequence, say (f. ) equivalent to the usual basis of or I * Xd q Let (y. ) denote the functionals in [f..]* biorthogonal to (f. ). ^ 10 *d

Since (f^) is an unconditional basic sequence, by Lemma 2 [8j,

(y ) is equivalent to (x. ) and hence (x ) is equivalent to the d d usual basis of Ag or as y^ is equivalent to the usual basis of .ftg or d Hence the lemma.

Remark

In the above lemma, the condition that [x^] be complemented is essential. In fact, Kadec and Pelczynski [8] have pointed out 35 that for 1 < p / q < 2 , there is a subspace of which is isomorphic to i .

Proof of Theorem 2 ,lk

Case I

Let p = 1, «. In this case, the hypothesis that no subspace of X is isomorphic to &g is unnecessary. In fact, by Lemma 1 . 2 5 and Theorem 1.26, (x^) Is equivalent to the usual basis of for p = and c for p = 1 o

Case II

First, without loss of generality, we can assume that X C ^ and complemented in L^ [Theorem 1.17]. By hypothesis and Lemma 2.15,

(x^) has a subsequence (x!p^) equivalent to the usual basis of X • j

Let X, = [x$l}]. Consider the sequence {(x.) \ ( x ^ } where \ ^ d denotes the set theoretic difference. Since (x^) is an unconditional basis and [x. 3 is complemented in L_, [(x.) \(xjU); )] is also comple- i p r Xj mented in 1^. Therefore, by Lemma 2.15, { ( x ^ V x ^(1 )- ')} has a d

subsequence (x^2 ^) equivalent to the usual basis of ^ . Let j yj*2^ =» xj[2* for d > 1 and yj:2 ^ = if [x£2 ^]. Then d J o d

X0 = [Xj2 ^, x. ] =5 [yj2 ^] is also equivalent to the usual basis 2 *d * d d=o of ip. Continuing inductively as above, we shall get a sequence (X^) of subspaces of X such that (1) X is spanned by a subsequence of (x.).

(2) X is isomorphic to £ . n p

(3 ) 0 Xj = {o }, i ^ j, and the natural projection of

n m X © X. -> S © X. j m < n has norm < X, 1 < i < nwhere i=i 1 i=i

X denotes the unconditional basis constant of (x^).

(V) TTx = X. n Clearly then X = £ © X where each X is isormqphic to I . Q.E. xi n p

Definition 2 . 1 6

Let X be a Banach space and A be a subspace of X.Then the projection constant of A with respect to X is the infimum ofthe norms of the projections from X onto A and is denoted by X^( a ).

Theorem 2.17

Let 1 < p < ®. Then the following conditions are equivalent:

(1) There exists a sequence of subspaces and a sequence (n^) of

integers such that

(n ) (i ) BBC£p »,d(Bm, ip ) < n for a fixed u and X- (B ) -» » as m -> ». jj m P (2) 3 an infinite dimensional subspace X of such that X ^ 4

but X is uncomplemented in 37

PROOF:

(1) - (2). (a ) Let Z = {Z ® 8>^m ) ^ , B|y [l6], Z is isomorphic to 1^. Let P

S denote the isomorphism of Z onto I . Clearly YCZ. Let sf

X = S(Y) c: j& . Since Y is isomorphic to i and S is an isomorphism, P P ve conclude that X is isomorphic to A • Assert that X is uncouple- Jr mented in A^. For this, it is enough to show that Y is uncomplemented in Z. Suppose Y is complemented in Z and P: Z -* Y is a projection.

Let P : Y -» B , m = 1,3,... be the natural projections and let m m (n ) (n ) = P/ip m , a -1,2,... . - V I '8' - 1

It is a straightforward calculation that R *s are projections and SI are uniformly bounded. Thus

( B j sup ( m m ) < m p which is a contradiction that X, (B ) « as m -* «>. This completes Xt SI p the proof that (l) ** (2 ).

Next we prove (2) =» (l).

Since X is isomorphic to A , it is a £ space for some I). P P j M (il ) Consider a finite dimensional sub space Bj c x such that d ( B ' , A_ ) X JL lr < T\. Since X is a £ space, 1 < p < °», there exists a finite dimensional P (io) subspace B' such that B* CB'CX, d(B’, A_ ) < 11 and X^ (B') > 2 wL ^ C ir p C

(otherwise, one can project onto every B', B* ^ B* 2 2 1 38 with norm < 2. But then Lemma 2.12 would imply that X is complemented in A X Inductively, we shall get a sequence (B*) of finite dimensional P n subspaces of X such that

(a) B'CB' Vn n n+1

(i > (b) a(B', i n ) < H n p

As a typical case, we show that we can get B ^ ' s as promised in the theorem. Let f_,...,f. denote a basis for B* end s > 0. «m n n n 3 an integer m 9 ||f - £ a e || < e, 1 < r < i where (e.) n * i^L ^ — ^ mn denotes the usual basis of A . Let g_ = £ a, e. . As in the p ^ i=i ir i proof of Proposition 2.1, we can show that g^’s are linearly independent and, in fact,

i i d ( [ f ] n , [g] n ) < 1 + e for a small e. r r=l ^ r=0.

i Let BQ = [gr] n . Then

d = d(V B’)d

< 1*1 + e) = H, say. 39 (m ) Also B c & n Vn, since there is a norm 1 projection from n p (m ) & onto & and X, (B') -* », we conclude that X. (B ) -> «*. *D J& IX * P P n p

Hence (2) ** (l), Q.E.D. CHAPTER III

SOME FINITE DIMENSIONAL SUBSPACE PROPERTIES

In this chapter, a property (the finite dimensional subspace basis property) which arises very naturally in the study of £ spaces and Jm in the Basis Theory is studied. Some other related properties are also studied.

Definition 3»1

A Banach space X has the X-finite dimensional subspace basis property

(written as \-f.d.s.b.) if there exists a constant X such that for any finite dimensional subspace B of X, there exists a finite dimen sional subspace W such that BCKX and T)(w) < X where T|(W) denotes the Infimum of the set of basis constants taken over all bases of W.

X is said to have f.d.s.b. if it has X-f.d.s.b. for some X.

Definition 3.2

A Banach space X is called a space if there exists a set

(B ) of finite dimensional subspaces of X, directed by inclusion, CL such that X = U B and there exists a projection P : X -> B , _ CL CL CL CL ||pj| < X for all a.

We have indirectly referred to the f.d.s.b. property in the 41

note at the end of Theorem 2.6. Since every Banach space with a

basis has this property (Cor. the property has interest from

the Basis Theory point of view.

Proposition 3.3

Let X be a Banach space such that X = U B_ where (B ) is an n

increasing sequence of finite dimensional subspaces of X. If there

exists a constant X such that

-f.d.s.b. for every e > 0. Conversely if X is separable and has

X-f.d.s.b., then there exists an increasing sequence (Bn ) of finite

dimensional subspaces such that X = U and iKB^) < X. n

PROOF:

Let anlr finite dimensional subspace of X and

e > 0. Then there exists an integer N and y^,y2 ,...,yn 6 B^ such

that j|x^ - yj| < 6 where 6 is arbitrary to be chosen suitably later.

As in the proof of Preposition 2.1, we can show that is a linearly independent set for a small 6. If we write IJjj = A, then w = e A

is well defined and, in fact, d(Bjj, W) < 1 + ^ by a suitably choice

of 6 (this can be achieved as in the proof of Proposition 2,1).

Since T|(bn) < X, we see that T|(w) < X(l + ~) = X + e. This completes the first part of the proposition. te

As for the second part, let (z^) be a dense set in X, Since

X has X-f.d.s.b., there exists a finite dimensional subspace such that z^ 6 and Tj(B^) < X. Similarly, there exists a finite dimensional subspace Bg such that [B^,Zg] c Bg and T|(Bg) < X.

Proceeding inductively, we get a sequence (B^) of finite dimensional subspaces of X such that

W Bn c Bn+1 *

(ii) Zn 6 Bn .

(iii) ^X, n = 1,2,3,.••

Thus X = U B . Q.E.D. n

Remark

Let X = U eT where (Bn ) is an increasing sequence of finite n dimensional subspaces. Suppose there exists a projection Pn : X -» Bn,

||P || < X, n =1,2,... where X is a constant. Then given B c X, dim(B) < “, there exists a finite dimensional sub space W, BCKX and a projection P: X -» W, ]|p !| < X + e for every e > 0. This result, for example, is discussed in [73*

Corollary 3.^

Let X be a Banach space with a basis (b^). Then X has (X + e)

-f.d.s.b. for every e > 0 where X is the basis constant for (b^). ^3

PROOF:

Let B = [ b . ] ? n. Then X = U B and T|(b ) < X. Therefore, n i x=u. _ n n — n the corollary follows from proposition 3* 3*

It is well known that every Banach space contains an infinite dimensional, subspace with a basis with basis constant less than or equal to 1 + e for every e > 0 [2], The following proposition shows that this result can be strengthened if the space has f.d.s.b. •

Proposition 3.5

Let X be a Banach space. Then the following conditions are equivalent:

(i) X has f.d.s.b.

(ii) there exists a constant n such that given any finite dimensional

subspace B cr X, there exists an infinite dimensional subspace Y

such that B cr y cr x, Y has a basis with basis constant less

than or equal to ji.

PROOF:

(i) ■» (ii). Let B cr X be any finite dimensional subspace. Let X have

X-f.d.s.b. • Therefore, 3 a finite dimensional subspace W such that

BCi/cx and TJ(w) < X. BY Lemma 2.7, there exists an infinite

dimensional subspace Z C X such that

(a) Z f! W ** (0). kk

equal to 1 + e for every e > 0.

There exists an infinite dimensional sub space Y' of Z with a basis (b^) and basis constant of (b^) less than or equal to 1 + e for every e > 0. Let Y « Y’ © W. Then the natural basis for Y has basis constant less than or equal to (l + c)(X + 2 + e). Hence

(i) =* (ii). That (ii) =* (i) is an immediate consequence of Corollary

3.^.

With the introduction of the f.d.s.b. property, quite a few natural questions arise e.g., if X has f.d.s.b., does X* have f.d.s.b. and conversely? In this direction, we have some partial results.

Proposition 3.6

If X** has X-f.d.s.b., then X has (X + e)-f.d.s.b. for every e > 0.

We need the following powerful result of Lindenstraxis s and

Rosenthal.

Lemma 3.7 [l1*]

Let X be a Banach space and e > 0. Let B c X** be a finite dimensional subspace. Then there exists a 1-1 operator T: B -»X such that T is identify on B H X and !|t ]| Ht *1!! < 1 + e.

Proof of Proposition 3*6

Let E c X be a finite dimensional subspace. Since X** has X-f.d.s.b., .*. there exists a finite dimensional subspace

B9ECJC X** and 11(b) < X. By Lemma 3.7 , there exists a 1-1 operator T: B -* X such that T is identity on B 0 X and

Mllof1!! < 1 + ^ . Then X 3 T(B) ^ E and

M b ) ) < x ||t || Ht’1 !! < x(i + £)

=* X + e

Q» E. D.

Proposition 3*8

Let be a Banach space such that X^ has X^-f.d.s.b., i ~ 1,2. Then ® Xg has f.d.s.b. .

PROOF:

Since f.d.s.b. is preserved under isomorphism, we can assume that

IKx^Xg)!! - llxjl + llxgH , xi € Xi# i = 1,2.

Let B c ® Xg be any finite dimensional subspace. 3 finite dimensional subspaces c X^, i a 1,2 such that BCE^® Eg. Also

3 c X^t d i m ^ ) < » such that llCW^) < X^, i » 1,2. Then

Proposition 3.9

Let a Banach space X have X- f. d. s. b. and Y X be a separable space. Then there exists a separable subspace Z € yczcX and Z has f.d.s.b. •

PROOF:

Le-fc (y )** , be a dense set in Y. Since X has X-f.d.s.b., 3 n n=tL a subspace B^ such that f c X and ll(B^) < X. Inductively, we shall have a sequence (Bn ) of finite dimensional subspaces such that

(i) [B^ yi+1J c Bi+1 cx,i= 1,2,...

(ii) T|(Bi ) < X, i = 1,2,...

Then, clearly, Z = U B is a separable subspace of X containing n n

Y. By Proposition 3.3, Z has (X + e)-f.d.s.b. for every e > 0. Q.E.D.

In the next proposition, we improve the above theorem if X Is reflexive.

Proposition 3.10

Let X be a reflexive Banach space with f.d.s.b. property. Let

Y c X be a separable subspace. Then 3 a separable space Z ^

(1) yczcx and Z has f.d.s.b.

(2) 3 a norm 1 projection from X onto Z.

We need the following result of Lindenstrauss. Lemma 3*11 [12]

Let X be a reflexive Banach space and W c X be a separable subspace. Then 3 a separable subspace L 3

(i) WCLCX,

(ii) 3 a norm 1 projection from X onto L.

Proof of Proposition 3.10

By Proposition 3.9, there exists a separable subspace Y1 3

(i) Y c Yx <= X.

(ii) Y^ has (X + e)-f.d.s.b. for every e > 0 .

By Lemma 3*11* there exists a separable sub space Yg 3

(iii) Y-l c Yg c x.

(iv) There exists a norm 1 projection from X onto Yg.

Continuing like this inductively, we get asequence (Y^) of separable subspaces of X such that

(a) Yfl G n = 1,2,...

(b) haS ^ + e)"ff o r every e > 0, n = 1 ,2 ,...

Let Z = U Y^. Then Y c z c X and Z is separable. Following the technique of Proposition 3.3, we can show that Z has f.d.s.b and thus (l) is satisfied. Also it can be shown, as in [Lemma 2, [12]], that through the sequence of projections (Pg^), we get a projection

P from X onto Z with ||p|| = 1. This proves (2). Q.E.D.

Theorem 3.12

Let X be a Banach space such that X does not have f.d.s.b. .

Then there exists a separable sub space Y c X such that

(i) Y does not have f.d.s.b.

(ii) Y has f.d.d.

We need the following lemmas:

Lemma

Let W be any finite codimensioneJL subspace of a Banach space

Z. If W has f.d.s.b., then Z has f.d.s.b.

PROOF:

Vie can write Z = W © [xi ]^ ^ for some linearly independent vectors, * * *,xn* 3 a real number \l such that ^ ^ i ^ i ^ - *1*

The frnJLowing lemma Is similar to Lemma 2* 10* Lemma 3.1**-

Let W be a finite codimensional subspace of a Banach space Z.

Suppose there exists a constant \ such that for any finite dimensional

subspace B <= w, there exists a finite dimensional subspace C such that

BCCCZ, T)( c ) < X , then Z has f.d.s.b. .

PROOF:

The proof will be given by induction on n, the number of codimension of W in Z.

Suppose Z = W © [x^ ]. Let B be a finite dimensional subspace of Z. There exists a finite dimensional subspace A C f f 3 B c :a © [x j ]»

By hypothesis of the lemma, 3 a finite dimensional subspace C 9

Acccz and Ti(c) < X. If x^ € C, then B c C c Z and we have already a 'nice' superspace of B. If C, let D = [C, x^] = C © [z^] for some z^ € Z. Then the ’nice' basis of C together with {z^lconstitutes a basis of D 9 T|(D) < 1 + 2X. Thus the result is proved for n » 1.

Suppose the result is known for codimension less than or equal to n-1. For n, let Z = W © Let C be any finite dimensional subspace of W. By the hypothesis of the lemma for n, 3 a finite dimensional subspace D such that C c D c z and T|(d) < X. If 1 D c w © [x^]^_^, then we proceed to the next step, otherwise let

Since 11(D) < X, therefore, 11(E) < 9*. Clearly C c E c \J & .

Thus we obtain that given any finite dimensional sub space C cff, 3 a finite dimensional subspace E 9 CCECff ® ^xi^i=& < 9^* 50

Thus the conditions for the validity for the induction hypothesis

for n-1 are satisfied and thus W 0 ^as • Therefore} by Proposition 3-8,

Proof of Theorem 3*12

Let (XQ ) be any increasing sequence of real numbers tending to infinity. Since X does not have f.d.s.b., given X^, there exists

a finite dimensional subspace B^cx such that for every finite dimensional subspace ^ ^ ^ c ^ c X> T|(W^) > X^. Choose one such

W1# By Lemma 2.7, given e > 0, there exists a finite codimensional

subspace Y^ of X such that

(i) Y^Y^-fO}.

(ii) the natural projection of Y^ ® *♦ has norm less than or

equal to 1 + e,

By Lemma 3.13, Y^ does not have f.d.s.b. . Therefore, given Xg ,

3 a finite dimensional sub space Bg of Y^ such that for any finite

dimensional subspace YL> such that Bg c wg c y ^ , T.(Wg) > Xg. By

Lemma 3.1^, we can choose one such that Wg c y , 5 ?,(wg) > Xg. By induction, using the technique of the proof of Theorem 2.6, we shall

get the sequences (Yn ) and (WQ ) such that 51

(i) Y„c y . y has finite codimension in Y ,, Y = X n n-1 n n-l o

(ii) Yn does not have have f.d.s.b.

(iii) tfn c Yn-1, dim(Wn ) < « and 'n(WJl) > \n *

has norm less than or equal to 1 + e, n = 1 ,2,...

(v) For any finite dimensional super space Kq of in

Then, as in the proof of Theorem 2.6, Y = Z © Wn is a well defined closed subspace of X. Clearly Y has f.d.d.so that (ii) is satisfied.

Also, from the construction of W 's, it is clear that W 's can not n n be contained in finite dimensional subspaces of Y with ’nice' basis constant and thus (i) is proved. Q.E.D.

Mow we introduce another property in Banachspaces viz. the

property.

Definition 3.15

A Banach space X is called a space if given any finite dimensional subsapce B cr X, there exists a finite dimensional subspace W such that B c W c X, tt:(w) < \ and there exists a projection

P: X *♦ W, ||p||< X.

Remark

The study of spaces arises naturally from spaces and also 5S the viewpoint of Basis Theory. Note that every space has

X-f.d.s.b. . The converse is unknown.

Example 3.16

(1) Every space with a basis is a B^ space. This follows from

Proposition 3.3 and the ramark at the end of Proposition 3*3*

(2) Every £ space is a B* space for some X. This follows frcan P «■ Theorem I.23.

In the following results, we study some natural questions associated with spaces such as if X* is a B^ space, is X a B^ space for some u? For these, we need two powerful lemmas proved recently by Johnson, Rosenthal and Zippin in [7 ].

We shall start with a definition.

Definition 3.17

Two Banach spaces X and Y are said to be eclose if there exists a 1-1 onto operator T: X -* Y such that ||t (x ) - x|| < e |]x||, x € X,

Theorem 3.18

If X* is a separable B. space, then X is a B space for some u. A. U

We need the following lemma:

Lemma 3.19 [Lemma 4.5 [7 ]]

Let X* be a it space represented as X = U E . Let E c x , A „ 0. Then there exists a finite rank projection Q, on X 9

(i) Q/E is identity.

(ii) M < 4X + h i ? .

(iii) Q*(X*) is e-dose to some Ea »

Proof of Theorem 3.18

Since X* is a separable space, there exists an increasing sequence (En ) of finite dimensional subspaces of X* such that

X* = U B , Tl(En) < X n and there exists a projection Pn : X* -» En , ||Pn || < X, n =* 1,2,... •

This can be achieved by taking a dense set in X* and then taking suitable finite dimensional superspaces as in the proof of Proposition

3.9. Let B c X be any finite dimensional subspace. By Lemma 3-19* there exists a finite rank projection Q on X such that <^B = 1 ^

Q*(X*) is e-close to some En , and j|o|| < 4X + 4x2, Since Tl(En ) < X, o o therefore,

•n.(Q**(x**)) < x j ? | I M

< \ (U\ + kxz).

But Q**(X**) * Q(X) =3 B.

Therefore, T!(Q(X)) < (ifX2 + *+X3)(|“ )

< 8(X2 + X3) If e < 3 * Q.E.D.

Remark

Since every space with a basis is a B space for some n, it follows immediately from Theorem 3*18 that if X* has a basis, then

X is a space for some X. It should be pointed out that much

stronger result is known viz. if X* has a basis, then X has a basis, C7l»

Definition 3*20

A Banach space X has n metric approximation property (written as n-m.a.p.) if given any finite dimensional subspace B C X and

e > 0, there exists an operator T on X, such that

||T|| < n a n d ||T(x) - x|| < e|lx|| V x € B.

Theorem 3.21

Let X be a separable B^ space and X* have n-m.a.p., then X* is a B -space for some r. r We need the following lemma:

Lemma 3*22 [Lemma [73]

Let X be a jt space represented as X = U E and X* have A. a _ O jx-m.a.p. • Let F c X* be a finite dimensional siibspace and e > 0. 55

Then there exists a finite rank projection Q on X such that Q*/F = I^,,

I!Oil < 2ti + 2X + tyiX and Q(X) iS e-close to some E . CL

Proof of Theorem 3.21

There exists an increasing sequence of finite dimensional subspaces of X such that

X = UEn 3 Tl(En) < \ n and there exists a projection pn ;X->E , l|Pnll<^j n=l,2,... . Let B C X * he any finite dimensional subspace. Then hy Lemma 3*22., there exists a finite rank projection Q on X such thatQ*(X*) B and Q(X) is e-close to some E . Since Tl(E ) < X, therefore, " o V

n(ft(x)) <

Therefore,

< X( ^ ~ )(2u. + 2X +VX)

< 2X(2n + 2X + VX) if e < i.

Q.E.D. 56

Remark

Theorems 3*18 and 3*21 are true without the condition of separability. The proofs are almost the same.

The proof of [Theorem 2.1, [l1*]] can be modified to give the following theorem:

Theorem 3*22

Let X be a space and Y be a complemented subspace of X.

Suppose there exists e > 0 such that every infinite dimensional subspace of X has a subspace e-isometric to X. Then Y has f.d.s.b. . CHATTER IV

SOME UNSOLVED PROBLEMS

In this chapter, we wish to summarize in a systematic way

some unsolved problems related to the material of this dissertation.

The first set of problems is related to £ spaces.

Problem 1

This problem was mentioned both in [133 and [I**-]. Theorem

2 . 1 1 and Theorem 2.lU give some partial answers to this problem.

As was pointed out before in Chapter II, the answer to Problem 1

is fno' for p s l, #,

Problem 2

Let 1 < p < », X be a Z space and X c l (\i) for seme measure P P H. Is it possible to find an embedding of X in L (n) such that the P corresponding embedded space is complemented in

Problem 2 is very naturally suggested by Problem 1. A positive answer to problem 2 would imply a positive answer to Problem 1, because if X c Jt and X is let Y be an embedding of X in ^ on which we can project. Then by a result of Pelczynski [l6], Y would be isomorphic to and hence X would be isomrophic to <8^. As was mentioned earlier in Chapter II, Problem 2 has a negative answer for p = 1, «*.

Problem 3

Let X, be isometric Banach spaces and each has infinite codimension in A . If X is complemented in A , is Y complemented P P in A , 1 < p < »? P “

Problem 4

In Theorem 2.6, can we choose Y with a basis?

The next set of problems is related to f.d.s.b. andproperties.

Problem 5

Does every Benach space have f.d.s.b.?

If there exists a Banach space X which does not have f.d.s.b., then by Theorem 3.12, X ^ Y such that Y does not have f.d.s.b. and

Y has f.d.d. . Therefore, by Proposition 3-3* Y can't have a basis.

Thus if the answer to Problem 5 is*no] then the basis problem would be answered negatively.

Problem 6

If X has f.d.s.b., does X* have f.d.s.b.?

Problem 7

If X* has f.d.s.b., does X have f.d.s.b.? 59

If the answer to Problem 6 is yes, then by Proposition 3*6, the answer to Problem 7 is also yes.

It is an unsolved problem that if a Banach space X has f.d.d., then it has a basis [3]. In connection with this we ask

Problem 8

If X has f.d.d. and f.d.s.b., does it have a basis? More strongly, if X has f.d.d. and it is a B^ space, does it have a basis?

If the answer to the question that X has f.d.d. =* X has a basis is yes, then by Theorem 3*12, it follows that every Banach space has f.d.s.b.

It is unknown that if a Banach space X has a basis, then every complemented subspace of X has a basis. We ask a formally weaker question.

Problem 9

If X has f.d.s.b., does every complemented subspace of X have f.d.s.b.? BIBLIOGRAPHY

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