Journal of Function Spaces

Journal of Function Spaces / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 9527954 | 8 pages | https://doi.org/10.1155/2019/9527954

Some Characterizations of -Strictly Convex and -Smooth Banach Spaces

Academic Editor: Alberto Fiorenza
Received05 Oct 2018
Accepted22 Jan 2019
Published06 Feb 2019

Abstract

In this paper, some characterizations of -strictly convex (-smooth, respectively) spaces are given. As special cases for , some characterizations of strictly convex (smooth, respectively) spaces are given.

1. Introduction

In 1936, Clarkson [1] introduced the concept of uniformly convex Banach spaces. Consequently, some methods were found to investigate the property of Banach space from the geometric structure of the unit sphere in Banach space. This initiated the study of convexity of Banach space. Since convexity has a striking intuitive geometric meaning, many mathematicians were attracted to this field of study. Smoothness, later introduced as a dual notion of convexity, is closely related to the various properties of differentiability of norm. This prompted further in-depth study of smoothness and the further development of the smooth theory of Banach space. As two important properties of geometry in Banach space, both convexity and smoothness play a very significant role in geometry of Banach space. They not only have promoted the development of geometric theories of Banach space, but also are also widely applied to such fields as control theory, operator theory, optimal approximation theory, and fixed-point theory.

In particular, the concept of a strictly convex Banach space is very interesting and has numerous applications (cf. [225], for example). In 1960, this concept has been generalized by Singer. He defined the so-called -strictly convex Banach space. The -strict convexity has important applications in approximation theory (cf. [12, 19, 2628], for example). It is well known that every -strictly convex space is -strictly convex. Also, it was shown that -strict convexity coincides with strict convexity. In order to describe spaces dual to those being -strictly convex space Nan and Wang introduced the notion of the so-called -smooth space. It is well known that every -smooth space is -smooth space. Also, it was shown that -smoothness coincide with smoothness.

The purpose of this paper is to give some necessary and sufficient conditions for -strict convexity (-smoothness, respectively).

Throughout the sequel, the symbol denotes a real Banach space and denotes its dual space. , denote the unit sphere in and the unit sphere in , respectively.

For each , set

For each , set

For each , set .

For each , set .

For , the -dimensional volume enclosed by is given by

Definition 1 (see [19]). A Banach space is said to be -strictly convex space provided that, for any norm-one elements in , if , then are linearly dependent.

Definition 2 (see [27]). A Banach space is said to be -smooth space provided that, for any norm-one element in , one has .

Lemma 3 (see [27]). If is -strictly convex, then is -smooth. If is -smooth, then is -strictly convex.

Lemma 4 (see [29]). Let be a Banach space. Suppose that are norm-one elements in and denotes the affine subspace generated by . Set

Then

2. Characterizations of - Strictly Convex Banach Spaces

Theorem 5. A Banach space is -strictly convex if and only if, for any norm-one elements in and real numbers with , the equality holds; then are linearly dependent.

Proof.
Necessity. Suppose the contrary; i.e., there are linearly independent norm-one elements of and real numbers satisfyingNow we are going to show that
If the above equality does not hold, then This yields this is impossible.
Combining with the assumption that is -strictly convex space, we know that are linearly dependent; this contradicts the fact that are linearly independent elements of .
Sufficiency. Suppose that, for any norm-one elements in , the following equality holds.
Let ; then we have From the conditions given here, we know that are linearly dependent. This shows that is a -strictly convex space.

Corollary 6. A Banach space is strictly convex if and only if, for any norm-one elements in and real numbers with , the equality holds; then are linearly dependent.

Theorem 7. A Banach space is -strictly convex if and only if, for any linearly independent norm-one elements in , real numbers with , and any functional , one has

Proof.
Necessity. Suppose that are any linearly independent norm-one elements in and are arbitrary real numbers with
Clearly, and the inequality holds for any functional
Now we are going to show that If the above inequality is not true, then Therefore,which leads to Hence By the -strict convexity of and Theorem 5, we know that are linearly dependent; hence we reach a contradiction.
Sufficiency. Suppose that are any norm-one elements of and Then, by Definition 1, it is necessary to show that are linearly dependent.
If are linearly independent, put , and then From the conditions given here, we know that the inequalityholds for any functional
On the other hand, by Hahn-Banach theorem, we know that there is a norm-one functional such that hence we reach a contradiction.

Corollary 8. A Banach space is strictly convex if and only if, for any linearly independent norm-one elements in , real numbers with , and any functional , one has

Theorem 9. Let be a Banach space and be any norm-one element of Then is -strictly convex if and only if, for any norm-one elements in linearly independent with , real numbers with , and any functional , one has

Proof.
Necessity. Suppose that are any norm-one elements of linearly independent with any norm-one element of and are arbitrary real numbers with . By the -strict convexity of and Theorem 7, we know thatHence, taking into account the fact that , we deriveSufficiency. If is not -strictly convex, then there are linearly independent norm-one elements of such that Let us take Then, by Hahn-Banach theorem we know that there is a norm-one functional such that This leads to Let , and then we have , but ; hence we reach a contradiction.

Corollary 10. Let be a Banach space and be any norm-one element of Then is strictly convex if and only if, for any norm-one element in linearly independent with , real numbers with , and any functional , one has

Theorem 11. A Banach space is -strictly convex if and only if, for any norm-one elements in and real numbers with , there is some functional such that and ; then are linearly dependent.

Proof.
Necessity. Let be any norm-one elements of and are arbitrary real numbers with Suppose that there is some functional such that and
Now we are going to prove that are linearly dependent.
If are linearly independent, then From , we haveit follows thatThis yields because of Hence, in view of the above real numbers with , we know that are linearly dependent from Theorem 5, a contradiction.
Sufficiency. Suppose that are any norm-one elements of and are arbitrary real numbers satisfyingNow we are going to prove that there is some functional such thatFirstly, we will prove that there is some functional such that
For with , by Hahn-Banach theorem we choose a norm-one functional such that ; i.e., Hence we have
Secondly, we will prove that
(i) If , then Hence, in view of the inequality and , we haveThis shows that
(ii) If , then ; it follows thatThis shows that
Combining (i) and (ii) we have
Hence, from the conditions given here, we know that are linearly dependent. So is -strictly convex space which follows from Theorem 5.

Corollary 12. A Banach space is -strictly convex if and only if, for any norm-one elements in and real numbers with , there is some functional such that and ; then are linearly dependent.

3. Characterizations of - Smooth Banach Spaces

Theorem 13. A Banach space is -smooth if and only if, for any linearly independent norm-one functionals , real numbers with , and any element , one has

Proof.
Necessity. Suppose that is a -smooth space. Then, by Definition 2, we have for any norm-one element of Suppose that are any linearly independent norm-one functionals of and are arbitrary real numbers with
Clearly, and the inequality holds for any element
Now we are going to show that
If the above inequality is not true, then we haveClearly, for every Combining the above equality with , we have .
Let , and then is a norm-one element of and This leads to , a contradiction.
Sufficiency. Suppose that for any linearly independent norm-one functions , real numbers with , and any element we have
If is not -smooth, then there is some norm-one element of such that This means that there are linearly independent norm-one functionals such that Put ; then , and , a contradiction.

Corollary 14. A Banach space is smooth if and only if for any linearly independent norm-one functionals of , real numbers with , and any element one has

Theorem 15. Let be a Banach space and be any norm-one functionals of which attains its norm on Then is -smooth if and only if, for any norm-one functionals linearly independent with , real numbers with , and any element , one has

Proof.
Necessity. Let . Suppose that are any norm-one functionals of linearly independent with and are arbitrary real numbers with . By the -smoothness of and Theorem 13, we know thatHence, taking into account the fact that , we deriveSufficiency. If is not -smooth, then there is some norm-one elements of such that This means that there are linearly independent norm-one functionals such that ; it follows that
Let , and then , but ; hence we reach a contradiction.

Corollary 16. Let be a Banach space and be any norm-one functionals of which attains its norm on Then is smooth if and only if, for any norm-one functional linearly independent with , real numbers with , and any element , one has

Theorem 17. A Banach space is -smooth if and only if, for any norm-one functionals and real numbers with , there is some element such that and ; then are linearly dependent.

Proof.
Necessity. Let be any norm-one functionals in and are any real numbers with Suppose that there is some element such that ThenFor , we have two possibilities: either , or
Case (I). If , then This means that are linearly dependent, so we obtain the desired result.
Case (II). If , we may assume that for Indeed, if some , we also obtain the desired result.
Let , and then is a norm-one functional of for and By Theorem 13, we know that are linearly dependent. Hence, are linearly dependent also, so we obtain the desired result.
Sufficiency. Suppose that is any norm-one element of and are any norm-one functionals of Then, for any real numbers with , we haveit follows that Hence we have
(i) If , thenHence, in view of the inequality and , we haveThis shows that
(ii) If , then ; it follows thatThis shows that
Combining (i) and (ii) we have
Hence, from the conditions given here, we know that are linearly dependent. This means that . It follows from Definition 2 that is -smooth space.

Corollary 18. A Banach space is smooth if and only if, for all norm-one functionals and real numbers with , there is some element such that and ; then are linearly dependent.

Theorem 19. A finite-dimensional Banach space is -smooth if and only if for any norm-one functionals and real numbers with , the equality holds; then are linearly dependent.

Proof.
Necessity. Suppose that are any norm-one functionals of and are arbitrary real numbers satisfyingSince is a finite-dimensional Banach space, so is reflexive. Therefore, there is a norm-one element in such thatit follows from thatThis means that By Definition 2 we know that are linearly dependent.
Sufficiency. It is immediate from Theorem 5 and Lemma 3.

Corollary 20. A finite-dimensional Banach is smooth if and only if, for any norm-one functionals and real numbers with , the equality holds; then are linearly dependent.

Theorem 21. A finite-dimensional Banach space is -smooth if and only if for any norm-one functionals of , real numbers with , and , one has

Proof.
Necessity. Suppose that are any norm-one functionals of and are arbitrary real numbers satisfyingBy Hahn-Banach theorem, we know that there is a norm-one functional such that By the assumption that , we have for every Since is a finite-dimensional Banach space, so is reflexive. Therefore, Furthermore, may be denoted by For every