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Abstract and Applied Analysis
Volume 2010, Article ID 102462, 14 pages
http://dx.doi.org/10.1155/2010/102462
Research Article

On - and -Convexity of Banach Spaces

Department of Mathematics, Center for Mathematical Reserch, A.C., Apdo. Postal 402, 36000 Guanajuato, GTO, Mexico

Received 25 June 2010; Accepted 18 August 2010

Academic Editor: Stevo Stevic

Copyright © 2010 Omar Muñiz-Pérez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show that every U-space and every Banach space X satisfying are P(3)-convex, and we study the nonuniform version of P-convexity, which we call p-convexity.

1. Introduction

Kottman introduced in 1970 the concept of P-convexity in [1]. He proved that every P-convex space is reflexive and also that P-convexity follows from uniform convexity, as well as from uniform smoothness. In this paper we study conditions which guarantee the P-convexity of a Banach space and generalize the result of Kottman concerning uniform convexity in two different ways: every -space and every Banach space satisfying are P()-convex. There are many convexity conditions of Banach spaces which have a uniform and also a nonuniform version, for example, strictly convexity is the nonuniform version of uniform convexity, smoothness is the nonuniform version of uniform smoothness, and a -space is the nonuniform version of a -space, among others. We also define the concept of -convexity, which is the nonuniform version of -convexity and obtain some interesting results.

2. -Convex Banach Spaces

Throughout this paper we adopt the following notation. will be a Banach space and when there is no possible confusion, we simply write . The unit ball and the unit sphere are denoted, respectively, by and . will denote the closed ball with center and radius . The topological dual space of is denoted by .

2.1. -Convexity

The next concept was given by Kottman in [1].

Definition 2.1. Let be a Banach space. For each let It is easy to see that for .

Definition 2.2. is said to be P-convex if for some .

The following lemma was proved in [1].

Lemma 2.3. Let be a Banach space and . Then if and only if there exists such that for any That is, is P -convex if and only if satisfies condition (2.2) for some and some .

Definition 2.4. Given and we say that is -convex if satisfies (2.2). For each , is said to be -convex if it is -convex for some .

2.2. -Convexity and the Coefficient of Convexity

In [1], Kottman proved that if is a Banach space satisfying the condition , then is P()-convex, where is the modulus of convexity. In this section we give a result which improves this condition, and we show that this assumption is sharp.

We recall the following concepts introduced by J. A. Clarkson in 1936.

Definition 2.5. The modulus of convexity of a Banach space is the function defined by The coefficient of convexity of a Banach space is the number defined as We also need the following definition given by R. C. James in 1964.

Definition 2.6. is said to be uniformly nonsquare if there exists such that for all In order to prove our theorem we need two known results which can be found in [2].

Lemma 2.7 (Goebel-Kirk). Let be a Banach space. For each , one has the equality .

Lemma 2.8 (Ullán). Let be a Banach space. For each the following inequality holds: .

Using these lemmas we obtain:

Theorem 2.9. Let be a Banach space which satisfies , that is, . Then is P()-convex. Moreover, there exists a Banach space with which is not P()-convex.

Proof. Let . Clearly . Let , and suppose that and . By Lemma 2.7, we have Similarly . Hence we get Finally, from Lemma 2.8 it follows that Then , and thus is P()-convex.
Now consider for each the space defined as follows. Each element may be represented as , where the respective th components of and are given by and . Set where stands for the -norm. The space satisfies (see [3]). On the other hand let , , , where is the canonical basis in . These points satisfy that , . Thus is not P(3.2)-convex.

It is known that if a Banach space satisfies , then has normal structure as well as P()-convexity. The space is an example of a Banach space with which does not have normal structure (see [3]) and is not P()-convex.

Kottman also proved in [1] that every uniformly smooth space is a P-convex space. We obtain a generalization of this fact. Before we show this result we recall the next concept.

Definition 2.10. The modulus of smoothness of a Banach space is the function defined by for each . is called uniformly smooth if .
The proofs of the following lemmas can be found in [4, 5].

Lemma 2.11. For every Banach space , one has .

Lemma 2.12. Let be a Banach space. is P()-convex if and only if is P()-convex.

By Theorem 2.9 and by the previous lemmas we deduce the next result.

Corollary 2.13. If is a Banach space satisfying , then is P()-convex.

With respect to P()-convex spaces we have this result, which is easy to prove.

Proposition 2.14. If is a Banach space -convex, then , and hence is uniformly nonsquare.

In fact, in bidimensional normed spaces, P()-convexity and uniform nonsquareness coincide. The proof of this involves many calculations and can be seen in [6].

Another technical proof (see [6]) shows that if is a bidimensional normed space, then is always P(1,5)-convex. Hence the space is P(1,5)-convex and , and thus P(5)-convexity does not imply uniform squareness.

2.3. Relation between -Spaces and -Convex Spaces

In this section we show that P-convexity follows from U-convexity. The following concept was introduced by Lau in 1978 [7].

Definition 2.15. A Banach space is called a U-space if for any there exists such that where for each
The modulus of this type of convexity was introduced by Gao in [8] and further studied by Mazcuñán-Navarro [9] and Saejung [10]. The following result is proved in [8].

Lemma 2.16. Let be a Banach space. If is U-space, then is uniformly nonsquare,

From the above we obtain the next theorem which is a generalization of Kottman's result, who showed in [1] that P()-convexity follows from uniform convexity.

Theorem 2.17. If is a U-space, then is P( )-convex.

Proof. By Lemma 2.16 we have that there exists such that for all Since is a U-space, for there exists such that We claim that is P -convex, where . Indeed, proceeding by contradiction, assume that there exist such that Define and , and let . If , then Therefore , which is not possible. Hence . Similarly we prove . Also , and hence, by (2.12) we have . By the above we have which is a contradiction.

2.4. The Dual Concept of -Convexity

In [1], Kottman introduces a property which turns out to be the dual concept of P-convexity. In this section we characterize the dual of a P-convex space in an easier way. We begin by showing Kottman's characterization.

Definition 2.18. Let be a Banach space and . A convex subset of is said to be -flat if . A collection of is called complemented if for each pair of and in we have that has a pair of antipodal points. For any we define

Theorem 2.19 (Kottman). Let be a Banach space and . Then (a)(b)

Now we define P-smoothness and prove that it turns out to be the dual concept of P-convexity. The advantage of this characterization is that it uses only simple concepts, and one does not need -flats. Besides in the proof of the duality we do not need Helly's theorem nor the theorem of Hahn-Banach, as Kottman does in Theorem 2.19.

Definition 2.20. Let be a Banach space and . For each set . Given and , is said to be -smooth if for each there exist , such that . is said to be -smooth if it is -smooth for some , and is said to be -smooth if it is -smooth for some and some .

Proposition 2.21. Let be a Banach space. Then
(a) is P()-convex if and only if is P()-smooth.(b) is P()-smooth if and only if is P()-convex.

Proof. (a) Let be a P()-convex space. Let . We will show that there exist , such that . Since is P-convex, it is also reflexive. Therefore for some , where 𝚥 is the canonical injection from to . By hypothesis, there exist , , such that . Therefore it is enough to prove that We proceed by contradiction supposing that there exists such that and . Then which is not possible; consequently is P()-smooth.
Now let be a Banach space such that is P()-smooth. Let . By hypothesis, there exist , , such that , that is, for each we have or . We will see that . We again proceed by contradiction supposing that . There exists such that If , then which is not possible. Similarly if , we obtain a contradiction. Thus , and consequently is P()-convex. The proof of (b) is analogous to the proof of (a).

Therefore the conditions is -smooth and must be equivalent.

3. -Convex Banach Spaces

In this section we introduce the nonuniform version of P-convexity and we call it p-convexity.

Definition 3.1. Let be a Banach space and . is said to be p -convex if for any , there exist , , such that . is said to be p-convex if is p()-convex for some .

Kottman defined the concept of P-convexity in terms of the intersection of balls. We will do something similar to give an equivalent definition of p-convexity. It is easy to see that in a normed space any two closed balls of radius contained in the unit ball have non empty intersection. If the radius is less than , for example, in for every and for every , then there exist closed balls of radius so that no two of them intersect. In fact let be the canonical basis of . Then the closed balls of radius centered at the points (, are disjoint and contained in the unit ball. However, if is p()-convex, we will see that for any points in the unit ball there exists so that if the closed balls centered at these points are contained in the unit ball, there are two different balls with non empty intersection. To prove this we need the following lemma, which was shown in [11].

Lemma 3.2. Let be a Banach space and , . Then

Lemma 3.3. is a p()-convex space if and only if for any there exists such that, if for all , then there are , , so that

Proof. Assume that satisfies condition (3.2), and let . Let be the number which satisfies condition (3.2) for . It is easy to see that for each . Therefore there exist , , such that Let We have and thus is p()-convex. Now we suppose that there exist such that for any we have for all , and for all , . We verify that is not p()-convex in four steps.(a)Take for any , .(b)Take, for all . To verify this claim we note that for all , because if for some , then , which is not possible. Hence, as , it follows that , for each . Now, if for some , we have by (a) that for any , which is not possible.(c)Take, for any , . Indeed, by (b) we get .(d)From (a), (b), (c), and by Lemma 3.2, we have for any , . Since is arbitrary, as , we obtain , for all , , and thus is not p(n)-convex.

Next we give some examples of spaces which are not p-convex. The first is not reflexive and the last one is superreflexive.

Example 3.4. , and consequently, and are not p-convex spaces. Indeed, let be the canonical basis in . For each we define , where if , and . Clearly and for each we have .

Example 3.5. Let denote the space obtained by renorming as follows. For set Then , where stands for the -norm and is superreflexive. On the other hand, the canonical basis in satisfies for each . Thus is not p-convex.

Now we will mention several properties that imply p-convexity.

Recall the following concepts. Let be a Banach space. is said to be a u-space if it satisfies the following implication: X is said to be smooth if for any , there exists a unique such that . That is, for each , contains a single point. X is called strictly convex if the following implication holds:

Proposition 3.6. Every smooth space, every strictly convex space and every u-space are p()-convex space.

Proof. Every smooth space and every strictly convex space are u-space. It suffices to show that p(-convexity follows from being u-space. If is a u-space, then for any the following inequality holds: . Indeed, if we suppose that there exist such that , then and , which is not possible. Suppose that is not p()-convex, and there exist so that . Since (we have . Let ; then , and Thus is p()-convex.

Obviously P-convexity implies p-convexity; however, a p-convex space is not necessarily P-convex, even if the space is reflexive as the following example shows.

Example 3.7. Let be a sequence of real numbers such that for each and , when . Consider the space . It is known that this space is strictly convex, hence it is also p( )-convex. It is also known that is reflexive. However is not P-convex. Indeed, let . We choose such that . If is the canonical basis of , we have that for all , , and hence is not a P-convex space.

We have obtained a result which shows a strong relation between P-convexity and p-convexity with respect to the ultrapower of Banach spaces. We recall the definition and some results regarding ultrapowers which can be found in [4].

A filter on is called an ultrafilter on if is a maximal element from with respect to the set inclusion. is an ultrafilter on if and only if for all either or . Let be a family of Banach spaces, and let If we define for each , then defines a norm in , and is a Banach space. If is a free ultrafilter on , then for each we have always exists and is unique. Let be an ultrafilter on , and define is a closed subspace of . The ultraproduct of with respect to the ultrafilter on is the quotient space equipped with the quotient norm, which is denoted by and its elements by . If for all , then is called the ultrapower of . The quotient norm in , satisfies the equality If is nontrivial, then can be embedded into isometrically. We will write instead of and instead of unless we need to specify the ultrafilter we are talking about.

It is known that is uniformly convex if and only if is strictly convex, is uniformly smooth if and only if is smooth, and is a U-space if and only if is a u-space (see [12]). Similarly we obtain the following result.

Theorem 3.8. Let be a Banach space and . The following are equivalent:

(a) is P()-convex.(b) is P()-convex,(c) is p()-convex,

Proof. . Let , , . Since for all , there exists a subsequence of such that and , for all . Define for each , . We verify that there exist , , such that . We proceed by contradiction assuming that, for all . Hence for all , and consequently therefore there exists . Thus we have for each , and is not P()-convex, which is a contradiction. Therefore there exist , , such that , and hence . Finally, note that Therefore is P()-convex.
is obvious.
. Suppose that is not P()-convex. Hence for any there exist such that for all , . Define for each . Clearly for all , because , and also, for each . Hence is not p()-convex.

By the above theorem we can deduce the following known result.

Corollary 3.9. If is P -convex, then is superreflexive.

Proof. If is P-convex, then is P-convex and therefore is reflexive. However in ultrapower reflexivity and superreflexivity are equivalent, hence is superreflexive, and consequently is superreflexive.

Now we turn our attention to some results regarding the p-convexity and the P-convexity of quotient spaces. To prove them we need the following concept.

Definition 3.10. A subspace of a normed space is said to be proximinal if for all there exists such that .

It is easy to see that every proximinal subspace of a Banach space is closed.

Proposition 3.11. If is p()-convex and is a proximinal subspace of , then is p ()-convex.

Proof. Let be the quotient function. By the proximinality of we have . Let and such that . Since is p()-convex, there exist , , such that , and consequently .

Corollary 3.12. Let be p()-convex and reflexive. If is a closed subspace of , then is p()-convex.

Proof. It is shown in [13] that a Banach space X is reflexive if and only if each closed subspace of is proximinal, and thus the corollary is a consequence of Proposition 3.11.

Similarly we can prove that if is P()-convex and is a closed subspace of , then is P()-convex.

We obtain two results involving -direct sums of p-convex spaces. Next we will define these sums as in [14] by Saito, et al.

Definition 3.13. Set
Let and be Banach spaces. For each , one defines the norm in as and for each

In [15] it is shown that is a Banach space, denoted by called the -direct and sum of and .

The proof of the following theorem is similar to the proof of Theorem in [16], which shows the corresponding result for P-convex spaces.

Theorem 3.14. Let and be Banach spaces and . Then is p-convex if and only if and are p-convex.

In [17] there is a theorem stating several equivalent conditions for strict convexity. We prove a similar result for p-convexity.

Lemma 3.15. Let be a Banach space. The next assertions are equivalent.(a) is p(n)-convex.(b)For any and for any , not all zero, there exist , , such that (c)For some and for any , not all zero, there exist , , such that

Proof. The implications are immediate. We verify . Let and , not all zero. If and for some , then it is clear that Suppose that . There exist , , such that If by Lemma 3.2 we get As the function is convex we obtain that Thus

Proposition 3.16. Let be a family of p(n)-convex spaces, where the index set has any cardinality. Then the space () is p(n)-convex.

Proof. Let , , not all zero. Let be such that , for some . As is a p(n)-convex space, we have by the preceding lemma that there exist such that By the above we obtain Therefore, by the previous lemma, is p(n)-convex.

Acknowledgments

The author is grateful to Helga Fetter and to Berta Gamboa for their valuable remarks and suggestions which led to substantial improvements in the paper. The author is partially supported by CONACyT, CIMAT, and SEP-CONACyT Grant 102380.

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