SYMMETRIC FINITE REPRESENTABILITY OF ℓP IN ORLICZ SPACES Текст научной статьи по специальности «Математика»

Submited: 14.10.2020 Revised: 16.11.2020 Accepted: 25.11.2020

S.V. Astashkin

Samara National Research University, Samara, Russian Federation E-mail: astash56@mail.ru. ORCID: https://orcid.org/0000-0002-8239-5661

SYMMETRIC FINITE REPRESENTABILITY OF ^ IN ORLICZ SPACES1

Scientific article

DOI: 10.18287/2541-7525-2020-26-4-15-24

ABSTRACT

It is well known that a Banach space need not contain any subspace isomorphic to a space lp (1 < p < to) or c0 (it was shown by Tsirel'son in 1974). At the same time, by the famous Krivine's theorem, every Banach space X always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of X of arbitrarily large dimension n which are isomorphic (uniformly) to i™ for some 1 ^ p < to or c$. In this case one says that lp (resp. c0) is finitely representable in X. The main purpose of this paper is to give a characterization (with a complete proof) of the set of p such that lp is symmetrically finitely representable in a separable Orlicz space.

Key words: lp-space; finite representability of lp-spaces; symmetric finite representability of lp-spaces; Orlicz function space; Orlicz sequence space; Matuszewska-Orlicz indices.

Citation. Astashkin S.V. Symmetric finite representability of lp in Orlicz spaces. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 2020, vol. 26, no. 4, pp. 15-24. DOI: http://doi.org/10.18287/2541-7525-2020-26-4-15-24.

Information about the conflict of interests: author and reviewers declare no conflict of interests.

© Astashkin S.V., 2020

Astashkin Sergey Vladimirovich — Doctor of Physical and Mathematical Sciences, professor, head of the Department of Functional Analysis and Function Theory, Samara National Research University, 34, Moskovskoye shosse, 443086, Russian Federation.

Introduction

While a Banach space X need not contain any subspace isomorphic to a space lp (1 < p < to) or c0 (as was shown by Tsirel'son in [1]), it will always contain at least one of these spaces locally. This means that there exist finite-dimensional subsets of X of arbitrarily large dimension n which are isomorphic (uniformly) to for some 1 < p < to or c$. This fact is the content of the famous result proved by Krivine in [2] (see also [3]). To state it we need some definitions.

Suppose X is a Banach space, 1 < p < to, and {zi}^=1 is a bounded sequence in X. The space lp is said to be block finitely representable in {zi}°=1 if for every n € N and e > 0 there exist 0 = m0 < m1 < ... < mn and ai € R such that the vectors uk = ^ ¿=°mk_1+1 aizi, k = 1, 2,.. .,n, satisfy the inequality

(1+ e)-1|M|

for arbitrary a

<

y^afc Uk

k=i

< (1 + £)№

X

(ak)n=i G Rn.

In what follows,

INI,

/ n := £1

k=i

\1/P

N M if p< œ, and ЦаЦœ :=

max

N |

The space lp, 1 < p < to, is said to be finitely representable in X if for every n € N and e > 0 there exist x1, x2,... ,xn € X such that for any a

(ak)n=i G Rn

(1+ £)-1MP <

E

k=i

ak Xk

< (1 + e)l|a||p

X

1The work was completed as a part of the implementation of the development program of the Scientific and Educational Mathematical Center the Volga Federal District, agreement no. 075-02-2021-1393.

(alternatively, in the case p = to, one might say that c0 is finitely representable in X).

Clearly, if lp is block finitely representable in some sequence {zi}°=1 C X, then lp is finitely representable in X. Therefore, the following famous result proved by Krivine in [2] (see also [3] and [4, Theorem 11.3.9]) implies the finite representability of lp for some 1 ^ p ^ to in any Banach space.

Theorem (Krivine)

Let {zibe an arbitrary normalized sequence in a Banach space X such that the vectors zi do not form a relatively compact set. Then lp is block finitely representable in {zi}°=1 for some p £ [1, to].

Here, we consider both Orlicz sequence and function spaces (see the next section for the definition) and in the separable case we give a characterization of the set of p such that lp is symmetrically finitely representable in such a space. To introduce the notion of symmetric finite representability, we need some more definitions.

A sequence y = (yk)kL1 will be called a copy of a sequence x = (xkif x and y have the same entries, that is, there is a permutation n of the set of positive integers such that yn(k) = xk for all k = 1, 2,....

Given a measurable function x(t) on [0,1], we set

nx(r) := m({t £ [0, a) : |x(t)| > t}), t > 0.

Here and in the sequel, m denotes the Lebesgue measure. Functions x(t) and y(t) are called equimeasurable if nx(T) = ny(t) for each t > 0.

Let X be a symmetric sequence space (see e.g. [5]), 1 < p < to. We say that lp is symmetrically finitely representable in X if for every n £ N and each e > 0 there exists an element x0 £ X such that for its disjoint copies xk, k = 1,2,...,n, and for every (ak)"=1 £ 1" we have

(1+ c)-1\\a\\p <

E<

au Xk

k=i

< (1 + z)H\p

X

Similar notion will be defined also in the function case. Let X be a symmetric function space on [0,1] [5]. The space lp is symmetrically finitely representable in X if for every n £ N and e > 0 there exist equimeasurable and disjointly supported on [0,1] functions ui(t), i = 1,2,... ,n, such that for all (ak )£=1 £ 1"

n

(1 - e)\\a\\p < || aiUi x < (1 + e)\\a\\p

i=i

X

The set of all p £ [1, to] such that lp is symmetrically finitely representable in X (in both sequence and function cases) we will denote by F(X).

From the definition2 of the Matuszewska-Orlicz indices aN and ¡3°N (resp. a^ and p^) of an Orlicz sequence space lN (resp. an Orlicz function space LN) it follows that F(X) C [aN] (resp. F(X) C [a^,p^]). The main purpose of this paper is to give a detailed proof of the opposite embedding for both Orlicz sequence and function spaces. To this end, following the idea mentioned in [6, p. 140-141] we will make use of the proof of Theorem 4.a.9 from [7].

Similar problems for Orlicz function spaces (and more generally symmetric spaces) on (0, to) were considered in the paper [8].

1. Preliminaries

1.1. Orlicz sequence spaces

A detailed information related to Orlicz sequence and function spaces see in monographs [9-11].

The Orlicz sequence spaces are a natural generalization of the lp-spaces, 1 < p < to, which equipped with the usual norms

.... J (Er=ik ip)1/p, 1 < P< to

\\a\\ep • ^ sup iaki , p = to. . [ fc=1,2,...

Let N be an Orlicz function, that is, an increasing convex continuous function on [0, to) such that N(0) = = 0 and lim^TO N(t) = to. The Orlicz sequence space lN consists of all sequences a = (ak)£=1, for which the following (Luxemburg) norm

\\a\\iN :=inJM> 0: £ N (^) < 1

I u=i

2See the next section.

is finite. Without loss of generality, we will assume that N(1) = 1. In particular, if N(t) = tp, we get the lp-space, 1 < p < to.

Recall that the Matuszewska-Orlicz indices (at zero) aN and ¡N of an Orlicz function N are defined by

0 { N(x)yp }

*N :=sup^ : ^ NX) < '

ßN := inf {p : inf

N(x)yp N(xy)

> 0} •

It can be easily checked that 1 < a0N < < to. It is well known also that an Orlicz sequence space lN is separable if and only if < to, or equivalently, if the function N satisfies the A2-condition at zero, i.e.,

N (2u) limsup < to.

u^o N (u)

The subset hN of an Orlicz sequence space lN consists of all (afc)fc=i € 1% such that

E n(

fc=i

Ы ^

u J

< то for each u >

One can easily check (see also [7, Proposition 4.a.2]) that hN is a separable closed subspace of and the canonical unit vectors en = (eln) such that en = 1 and eln =0 if i = n, n =1, 2,..., form a symmetric basis of the space hN. Recall that a basis {xnof a Banach space X is said to be symmetric if there exists C > 0 such that for any permutation n of the set of positive integers and all an G R we have

C-

IE-

n=1

<

X

IE-

n=1

nXn(n)

<

X

с ||E'

n=1

X

Observe that the definition of an Orlicz sequence space lN is determined (up to equivalence of norms) by the behaviour of the function N near zero. More precisely, the following conditions are equivalent: 1) lN = lM (with equivalence of norms); 2) the canonical vector bases of the spaces hN h hM are equivalent; 3) there are constants C > 0, c > 0 and t0 > 0 such that for all 0 < t < t0 it holds

cN(C-11) < M(t) < c-1N(Ct)

(see e.g. [7, Proposition 4.a.5] or [11, Theorem 3.4]). In particular, if N is a degenerate Orlicz function, i. e., for some t0 > 0 we have N(t) = 0 if 0 < t < t0, then lN = l(with equivalence of norms). Given Orlicz function N, we define the following subsets of the space C[0, 2]:

eN ,<

r N (xy) 1 N (y)

: 0 <y < a},

eN

П

0<a<1

eN ,î

and

0

CN,n

convEN (

0

CN

n

0<a< 1

0

CN,i

where 0 < a < 1 and the closure is taken in the norm topology of C [0,1 ]. All these sets are non-void norm compact subsets of the space C[0,1 ] [7, Lemma 4.a.6]. It is well known that they determine to a large extent the structure of disjoint sequences of Orlicz sequence spaces (see [7, §4.a] and [12]). Moreover, if 1 < p < to, then tp € CN if and only if p e [aN, ¡N] [7, Theorem 4.a.9].

In the case when an Orlicz function N satisfies the A2-condition at zero, the sets E0

N,a-

EN,

C0 can be considered as subsets of the space C[0,1] (see the remark after Lemma 4.a.6 in [7]).

CN,a and

1

n^n

n^n

1.2. Orlicz function spaces

Let N be an Orlicz function such that N(1) = 1. Denote by LN the Orlicz space on [0,1] endowed with the Luxemburg norm

1

......... ........'\x(t)\\

N

1

||x||L„ := inf{u> 0: J n(dt < 1}

In particular, if N(t) = tp, 1 < p < to, we obtain the space Lp = Lp[0,1] with the usual norm.

The Matuszewska-Orlicz indices a^ and (at infinity) of an Orlicz function N are defined by the formulae

^ r N (x)yp } ^ r N (x)yp }

a»=su^p: s NX) < , =inf {p: i Nxy) > 0}.

Again 1 < a^ < ¡3^ < œ. As in the case of sequence spaces, an Orlicz space LN is separable if and only if < œ, or equivalently, if the function N satisfies the A2-condition at infinity, i.e.,

N(2u) umsup < œ.

U^^ N (u)

In contrast to the sequence case, the definition of an Orlicz function space LN on [0,1] is determined (up to equivalence of norms) by the behaviour of the function N(t) for large values of t. For every Orlicz function N we define the following subsets of the space C[0, ^]:

EA := {Ny : y>A),E^ = П a, := convE

œ N ,

A>0

where the closure is taken in the norm topology of C[0, ^]. Again all these sets are non-void norm compact subsets of the space C[0,1 ] and they determine largely the structure of disjoint sequences in Orlicz function spaces (see [12, Propositions 3 and 4]). Moreover, if 1 < p < to, then tp € C^ if and only if p € [12].

Finally, if an Orlicz function N satisfies the A2-condition at infinity, the sets E^ A, E^ and C^ can be considered as subsets of the space C[0,1].

2. Symmetric finite representability of lp in Orlicz sequence spaces

Theorem 1

Let M be an Orlicz function satisfying the A2-condition at zero. Then lp is symmetrically finitely representable in the Orlicz sequence space lM if and only if p € [a°M,j°M], i.e., F(lM) = [a0M,jM].

Proof.

As was observed in §1, we always have F(lM) C [aM, jM]. Therefore, it suffices to prove only the opposite embedding. In other words, we need to show that for every p € [a°M,j°M], m € N and each e > 0 there exists an element x0 € lM such that for its disjoint copies xk, k = 1, 2,... ,m, and for every c = (ck)m=1 € Rn we have

(1 + c)-1II4p <

y^Ck^k

k=1

< (1+ ¿IIP (1)

Im

According to the proof of Theorem 4.a.9 in [7] and a comment followed this proof on p. 144, tp € C

0

M

(see also §2.1). Since M satisfies the A2-condition at zero, the set CM may be considered as a subset of the space C[0,1] (see the remark after Lemma 4.a.6 in [7] or again §2.1). Therefore, since CM := Pi0<o<1 C%ia,

0

M

Note that the mapping

we conclude that tp G CM 2_„ for each n G N.

Л ^ Mx(t) := M (At)/M (A) (2)

rom In := (0, 2-n] into th

Theorem 1.1]),

is continuous from In := (0, 2 n] into the subset E°M 2-n of C[0,1]. Indeed, as it is well known (see e.g. [9,

0

where p is a nondecreasing right-continuous function.

Therefore, for arbitrary A2 > A1 > 0 and all 0 < t < 1 we have

M (t) = i p(s) ds, (3)

0

IMX2 (t) - MXl (t)|

= IM(Ai)M(A2t) - M(A2)M(Ait)I = M(Ai)M(A2)

< MA)(M (A2t) - M (Ai t) + M (A2) - M (Ai))

1 (

<

£ p(s) ds+/Jp(s) dss

< 2P(A2) (A A )

< MA)(A2 - Ai).

Thus, mapping (2) may be extended uniquely to a map w ^ Mu from the Stone-Cech compactification jjIn of In onto the set E°M 2_„_i. Since tp € CM 2-n and the extreme points of CM 2-n are contained in the

compact set E°M 2-n, by the Krein-Milman theorem (see e.g. [13, Theorem 3.28]), there exists a probability measure nn on the set 3In such that

i Иш(t) d^(u), 0 С t С 1. Jein

(4)

Let us show that

for some probability measure vn on In we have

Mx(t) dvn(X)

< 2-n, 0 С t С 1.

(5)

First, the fact that the set Qn := Q n In (Q is the set of rationals) is dense in ¡3In implies that the set {Mr,r e Qn} is dense in the subset [Mue 3In} of C[0,1]. Consequently, putting Qn = {rkand

Ek := [u £ /3In : \MU (t) - Mrk (t)| < 2-n for all 0 С t С 1},

(6)

we have ¡3In = |JEk. Now, if Fm := Em \ (Um=1 Ek), m = 1, 2,..., then Fm are pairwise disjoint and ¡3In = UW=1Fm. Define the measure vn on a-algebra of Borel subsets U of the interval In by

Vn(U) := UnF),

{k: rkeu}

where nn is the probability measure from (4). Since

w

Vn (In ) = ^2 Un(Fk ) = VnWIn) = 1,

k=1

then vn is a probability measure on In. Moreover, by (4) and (6), for all 0 < t < 1

M\(t) dvn(X)

( Mu(t) d^n(u) - f Mx(t) dvn(X) J ein Jo

С El / Mu (t) d^n(u) - Mx(t) dvn (A) k=l Jpk J{rk}

со л p

С El/ (Ma (t) + 2-n) dvn(A) - Mx (t) dvn(A)

1--л J-irh\ J-irk 1

k=l ' J{rk}

ТО

n

С 2-nJ2 Vn({rk}) = 2-nVn(In) =2

nl

k=l

and inequality (5) is proved.

Next, for any s e (0,1) and n,j e N we set

sj-12-n

j, n •

I sj 2-n

dvn(A)

M(A) .

Then, by inequality (5), we have

J2[aj,n]M(sj2-nt) - 2-n <tp <J2laj,n]M(sj-l2-nt) + M(t)2-n/(1 - s) + 2

j=i

j=i

where by [z] we denote the integer part of a real number z. Choosing now kn such that

w

]T a,n]M(sj-12-n) < 2-n,

j=kn+1

as M(t) < M(1) = 1, we get

Fn(st) - 2-n+1 <tp < Fn(t) + 2-n/(1 - s) + 2-n+1, 0 < t < 1,

where

kn

Fn(t) :=^2[ajtn]M(sj-12-nt). j=1

(7)

(8) (9)

2

0

2

0

n

Since the right derivative p of M (see (3)) is a nondecreasing function and 0 < s < 1, from (7) it follows that

Fn(t) - Fn(st) * ]T ajn(M(sj-12-nt) - M(sj2-nt))

j=i

2-nsj-1(1 - sWsj-12-n) fs'-l2-n 7 ,л.

* g--1,2-n

Furthermore, the estimate

f2x

i'2x

F(2x) > / p(s) ds > xp(x), 0 < x < 1,

J x

combined with the hypothesis that M satisfies the A2-condition at zero, shows that

xp(x)

K := sup < to.

0<x^i M (x)

Hence,

Fn(t) - Fn(St) < K (1 - S) g M (Sj 2-n) J, - d»n(A)

Moreover, one can readily check that the upper Matuszewska-Orlicz index j M0 is finite (see also § 2.1) and, by its definition, for each q > jjM there is a constant c0 > 0 such that

M(sj2-n) > c0M(sj-12-n)sq.

As a result, since vn is a probability measure, we conclude

kn rsj-12-n

/>S" 2

Fn(t) - Fn(st) < K(1 - s)s-qc-1^ dvn(A) < K(1 - s)s-qc-1. (10)

j = 1J sj 2-n

Let m € N and e > 0 be arbitrary. Choose and fix s € (0,1) so that

K(1 - s)s-qc-1 <e/(2m). (11)

Then, from (8) and (10) it follows

Fn(t) - — - 2-n+1 < Fn(st) - 2-n+1 <tp, 0 < t < 1. (12)

2m

Now, taking n € N satisfying the inequality

2-n e

+ 2-n+1 < — (13)

1 - s 2m

from (8) and (12), we obtain

F F

Fn(t) - - <tp < Fn(t) + - 0 < t < 1. (14)

mm

Therefore, for any ci € [0,1], i = 1, 2,... ,m,

m m m

J^cp - e<J^Fn(ci) <J2cp + i=1 i=1 i=1

whence for all c = (ck)n=i € Rn, ck ^ 0,

1 - e< t it) <'+-

Moreover, since Fn is a convex function, from the latter inequality it follows that

m f \ g Fn{ (ITFMP) * 1

and

m .

I

,(1 - f)\\C\\

m / N

g Fn{ (1 -CF)\\c\\p) > ^

Therefore, by the definition of the norm in an Orlicz sequence space, for every m e N and all c = (ck)n=1 e Rn we have

m

(1 - e)W4p < II £ ciei < (1 + e)\\c\\p, (15)

11 i=1 lFn

where ei, i = 1, 2,..., are the canonical unit vectors in lFn.

Given m e N and e > 0, select s and n to satisfy (11) and (13). For any i = 1, 2,... ,m and j = 1, 2,... ,kn denote by Aj, n pairwise disjoint subsets of positive integers such that card Aj, n = [aj,n]. Then, the vectors

kn

ui :=2-n^ sj-1 Y^ ek, i = 1, 2,...,m, j=1 keAjn

are copies of an element from lm. Moreover, by formula (9), we have

mm

II / , ciui || / ,ciei

11 = £M 11 i=1 F

for all ci e R. Combining this with (15), we get (1), which completes the proof.

3. Symmetric finite representability of lp in Orlicz function spaces

Theorem 2

Let M be an Orlicz function satisfying A2-condition at infinity. Then lp is symmetrically finitely representable in the Orlicz function space LM if and only if p e [aM,3m], i.e., F(LM) = [aM,3m].

Proof.

As in the sequence case, we need only to prove the embedding [aW,3m] C F(Lm). More precisely, we have to check that for every p e [aM,3m], m e N and each e > 0 there exist equimeasurable and disjointly supported functions uk, k = 1, 2,...,m, satisfying for all c = (ck )m=1 e Rm the inequality:

(1 + c)-l\\c\\p С

С (1 + £)\\c\\p (16)

p LM

ck uk I

k=1

First, tp e CM C C[0,1] and then the same reasoning as in the proof of Theorem 1 shows that and that for every n e N there is a probabilistic measure vn on [2n, to) such that for all t e [0,1]

M (At)

лСО

/2П M(A) dvn(A)

< 2-n.

For any s > 1 and n, j e N we define

= fs 2n d^n(A)

aj,n : Jsj-I2n M(A) .

Then, by the preceding inequality,

CO CO

J^aj,nM(sj-12nt) - 2-n <tp <J^aj,nM(sj2nt) + 2-n. j=1 j=1

Next, as M satisfies the A2-condition at infinity, we have

M(sj2nt) < (1 + 2-n)M(sj-12nt)

for all j e N and t e [0,1] whenever s is sufficiently close to 1. Fixing such a s, we get

oo oo

Y,aj,nM (sj-12nt) - 2-n <tp <J2 (1 + 2-n )ajnM (sj-12nt) + 2-n. j=1 j=1

Combining this inequality with the estimate

2-n J2 ajnM(sj-l2nt) < 2-2n + 2-ntp < 2-n+l, 0 С t С 1, j=l

we deduce

oo oo

Y,aj,nM(sj-l2nt) - 2-n <tp <Y,aj,nM(sj-l2nt) + 2-n+2. (17)

j=l j=l

On the other hand, since M(u) ^ u for all u ^ 1, we have

which implies that

a, n S —:———:—T S 2"ns"j+1, j ' M (2nsJ-1 '

, n S 2"n£ s-j+1 = 2"n+ . .

j=i j=i

Let m £ N and e > 0 be arbitrary. Fix n so that

o-n+1„ 1

-— < — and 2-n+2m < e. (18)

s — 1 m

The first of the inequalities (18) allows us to take pairwise disjoint sets Ej c [0,1], j £ N, i = 1, 2,...,m, with m(Ej) = aj,n. Then, the functions

œ

ui := Y,2nsj"1XE) j=i

are equimeasurable and disjointly supported on [0,1]. Moreover, for all Ci G R

f.1 m m œ

V CiUi (t) I dt ^У M 12" S -ICil )a

■i ¡J ^j, n •

M (I E CiUi (t)|) dt = EE M (2V-1C "w i=1 i=1j=1

Therefore, by (17) and the second inequality in (18), we get

m 1m m

Y.\ci\p - £< M CiUi(t)) dt<J2 |Ci|p + £•

= 1 1/0 i=l i=1 Repeating further the arguments from the end of the proof of Theorem 1, we come to (16) and so complete the proof.

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[12] Lindenstrauss Y., Tzafriri L. On Orlicz sequence spaces. III. Israel Journal of Mathematics, 1973, vol. 14, pp. 368-389. DOI: https://doi.org/10.1007/BF02771656.

[13] Rudin W. Functional Analysis. Moscow: Mir, 1975, 443 p. Available at: https://www.nehudlit.ru/books/ funktsionalnyy-analiz.html.

Научная статья DOI: 10.18287/2541-7525-2020-26-4-15-24

УДК 517.982.27 Дата: поступления статьи: 14.10.2020

после рецензирования: 16.11.2020 принятия статьи: 25.11.2020

С.В. Асташкин

Самарский национальный исследовательский университет имени академика С.П. Королева, г. Самара, Российская Федерация E-mail: astash56@mail.ru. ORCID: https://orcid.org/0000-0002-8239-5661

СИММЕТРИЧНАЯ ФИНИТНАЯ ПРЕДСТАВИМОСТЬ Р В ПРОСТРАНСТВАХ ОРЛИЧА3

АННОТАЦИЯ

Хорошо известно, что банахово пространство может не содержать подпространств, изоморфных хотя бы одному из пространств lp (1 ^ p < ж) или с0 (это было показано Цирельсоном в 1974 г.). В то же время по известной теореме Кривина каждое банахово пространство X всегда содержит хотя бы одно из этих пространств локально, т. е. существуют конечномерные подпространства в X сколь угодно большой размерности n, изоморфны (равномерно) для некоторых 1 ^ p < ж или сЩ. В этом случае говорят, что lp (соответственно с0) финитно представимо в X. Основная цель этой статьи — дать характеризацию (с полным доказательством) множества тех p, что lp симметрично финитно представимо в любом сепарабельном пространстве Орлича.

Ключевые слова: ¿^-пространство; финитная представимость ¿^пространств; симметричная финитная представимость ¿^пространств; функциональное пространство Орлича; пространство последовательностей Орлича; индексы Матушевской — Орлича.

Цитирование. Astashkin S.V. Symmetric finite representability of lp in Orlicz spaces // Вестник Самарского университета. Естественнонаучная серия. 2020. Т. 26, № 4. С. 15-24. DOI: http://doi.org/10.18287/2541-7525-2020-26-4-15-24.

Информация о конфликте интересов: автор и рецензенты заявляют об отсутствии конфликта интересов.

© Асташкин С.В., 2020

Асташкин Сергей Владимирович — доктор физико-математических наук, профессор, заведующий кафедрой функционального анализа и теории функций, Самарский национальный исследовательский университет имени академика С.П. Королева, 443086, Российская Федерация, г. Самара, Московское шоссе, 34.

Литература

[1] Цирельсон Б.С. Не в любое банахово пространство можно вложить lp или с0 // Функц. анал. и его прил. 1974. Т. 8, №2. С. 57-60. URL: http://mi.mathnet.ru/faa2331.

[2] Krivine J.L. Sous-espaces de dimension finie des espaces de Banach reticules // Annals of Mathematics. 1976. Vol. 104, № 2. P. 1-29. URL: https://www.irif.fr/ krivine/articles/Espaces_reticules.pdf.

3Работа выполнена в рамках внедрения программы развития Научно-образовательного математического центра Приволжского федерального округа, договор № 075-02-2021-1393.

[3] Rosenthal H.P. On a theorem of J.L. Krivine concerning block finite representability of lv in general Banach spaces // Journal of Functional Analysis. 1978. Vol. 28. P. 197-225. DOI: http://dx.doi.org/10.1016/0022-1236(78)90086-1.

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[5] Крейн С.Г., Петунин Ю.И., Семенов Е.М. Интерполяция линейных операторов. Москва: Наука, 1978. 400 с. URL: https://elibrary.ru/item.asp?id=21722209; https://booksee.org/book/577975.

[6] Lindenstrauss J., Tzafriri L. Classical Banach Spaces, II. Function Spaces. Berlin, Heidelberg, New York: Springer-Verlag, 1979. 243 p. URL: https://1lib.education/book/2307307/8b833b?dsource=recommend.

[7] Lindenstrauss J., Tzafriri L. Classical Banach Spaces, I. Sequence Spaces. Berlin-New York: Springer-Verlag, 1977. 190 p. URL: https://1lib.education/book/2264754/01841c?dsource=recommend.

[8] Асташкин С.В. О финитной представимости 1р-пространств в перестановочно инвариантных пространствах // Алгебра и анализ. 2011. Т. 23, № 2. С. 77-101. URL: http://mi.mathnet.ru/aa1235

[9] Красносельский М.А., Рутицкий Я.Б. Выпуклые функции и пространства Орлича. Москва: Гос. изд-во физ.-мат. лит., 1958. 271 с. URL: https://1lib.education/book/2078048/983381?id=2078048&secret=983381.

[10] Rao M.M., Ren Z.D. Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146 - Marcel Dekker Inc., New York, 1991. 445 p.

[11] Maligranda L. Orlicz Spaces and Interpolation. Seminars in Mathematics 5. Campinas: University of Campinas, 1989. 206 p.

[12] Lindenstrauss Y., Tzafriri L. On Orlicz sequence spaces. III // Israel Journal of Mathematics. 1973. Vol. 14. P. 368-389. DOI: https://doi.org/10.1007/BF02771656.

[13] Рудин У. Функциональный анализ. Москва: Мир, 1975. 443 с. URL: https://www.nehudlit.ru/books/ funktsionalnyy-analiz.html.