New Characterizations of -Uniformly Extremely Convex Banach Spaces

Abstract

Two characterizations of -uniformly extremely convex spaces are given in this paper. As special cases for , two necessary and sufficient conditions of uniform extreme convexity are presented and a characteristic inequality of uniformly extremely convex spaces is given.

1. Introduction

In the geometric theory of Banach spaces, the concept of uniform convexity plays a very significant role and is frequently used in functional analysis. In 2012, this concept has been generalized by Wulede and Ha [1]. They defined the so-called uniform extreme convexity and showed many interesting connections between this concept and several others (see [1–8]). Recently, Wulede et al. [9], generalizing the uniform extreme convexity, introduced the so-called -uniform extreme convexity. Every -uniformly extremely convex space is -uniformly extremely convex, but the converse implication is not true (see [9]). It is shown that -uniform extreme convexity coincides with uniform extreme convexity (see [9]).

In 1960, Singer [10] introduced -strictly convex spaces. It is well known that every -strictly convex space is -strictly convex space. Also, it was shown that -strict convexity coincides with strict convexity.

Throughout the sequel, the symbol denotes a real Banach space and denotes its dual space. , denote the closed unit ball and the unit sphere in , respectively. For , the -dimensional volume enclosed by is given by

In 1972, Daneš [11] proved the so-called “Drop Theorem”. In 1987, modifying the assumption of Daneš drop theorem, Rolewicz [6] began the study of drop property for the closed unit ball. He defined the norm to have the drop property, if for every closed set disjoint from closed unit ball there exists a such that , where the set , the convex hull of and , is called the drop generated by .

Definition 1 (see [1]). A Banach space is said to be uniformly extremely convex space provided that for any sequences of norm-one elements of , if there is a norm-one functional such that , then

Definition 2 (see [9]). A Banach space is said to be -uniformly extremely convex space provided that for any sequences of norm-one elements of , if there is a norm-one functional such that for , then

Lemma 3 (see [9]). A Banach space X is -uniformly extremely convex if and only if for any there exists such that for each norm-one functional and for all norm-one elements of with , one has .

Lemma 4 (Singer [10], Wulede [9]). Every strictly convex (uniformly extremely convex, respectively) space is -strictly convex (-uniformly extremely convex, respectively).

Lemma 5 (Wulede [1, 9]). Let be a Banach space; let be an integer with . Then, is -uniformly extremely convex if and only if is -strictly convex and has the drop property.

Lemma 6 (see [12]). Let ; then we havewhere , and the sign of equality holds if and only if

2. Characterizations of -Uniformly Extremely Convex Spaces

Theorem 7. A Banach space is -uniformly extremely convex if and only if for any sequences , if and for some norm-one functional , , then

Proof. ⁡
Necessity. Without loss of generality, we may assume that and suppose that for any sequences with , and for some norm-one functional ,
Let , and then we have , andIt follows that By the assumption that is a -uniformly extremely convex space, we have Hence,It follows that
Sufficiency. Suppose that are sequences of norm-one elements of such that for some norm-one functional the equality holds for It follows that From the conditions given here, we haveThis shows that is -uniformly extremely convex space.

Theorem 8. Let , be a Banach space, and be an arbitrary bounded subset of . Then is -uniformly extremely convex if and only if for any there exists such that the inequalityholds for any norm-one functional and all elements with

Proof. ⁡
Sufficiency. Suppose that for every there is a such that the inequalityholds for any norm-one functional and all norm-one elements of with
Let us denote and take . Then we haveBy Lemma 3, we know that is a -uniformly extremely convex space.
Necessity. Suppose the inequality (6) is not true. Then there exist and such that for each N there exist satisfying for , andPut and take Rearranging the vectors, we can assume that We have for every andCombining inequality (10) with Lemma 6 we derive thatIt follows thatWe shall show thatfor every . To this end, it is enough to show that every subsequence of the sequence has a further subsequence converging to 1. Since every bounded sequence has a convergent subsequence, without loss of generality, we may assume that the sequence converges to some . The right equality in (13) shows thatwhere This is the equality in Hlder’s inequality, so it holds only iffor every Consequently, for every and the left equality in (13) shows thatCombining equality (17) with the assumption that is a -uniformly extremely convex space, by Theorem 7 we havethis contradicts (11).

In particular, considering the special cases of Theorems 7 and 8 when k = 1, we obtain a necessary and sufficient condition for uniform extreme convexity and a characteristic inequality of uniformly extremely convex spaces.

Corollary 9. A Banach space is uniformly extremely convex if and only if for any sequences in , if   and for some norm-one functional , , then

Corollary 10. Let , be a Banach space, and be an arbitrary bounded subset of . Then is uniformly extremely convex if and only if for any , there exists such that the inequalityholds for any norm-one functional and all elements with .

In addition to two characterizations of uniformly extremely convex spaces presented in Lemmas 4 and 5, we can also obtain another feature of uniformly extremely convex spaces as follows.

Corollary 11. A Banach space is uniformly extremely convex if and only if is strictly convex and -uniformly extremely convex.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11561053).