Characterization of Reflexivity by Convex Functions

Like differentiability, convexity of functions has proven to be a very useful tool to characterize various classes of Banach spaces. We have also known that good progress has been made in this direction. See, for instance, [1–10]. In particular, Zălinescu [10] first proved that a Banach space is reflexive provided that there exists a continuous uniformly convex function on some nonempty open convex subset of the space. Later, Cheng et al. [11] proved that, in fact, such a Banach space is super reflexive, and vice versa (recall that a Banach space is super reflexive if and only if it admits an equivalent uniformly convex norm [12]). Recently, this result has also been obtained independently by Borwein et al. (see [2, Theorem 2.4]). In particular, in [2] they mainly investigated the relationship between the existence of uniformly convex functions f : X → R bounded above by some power type of the norm and the existence of an equivalent norm with a certain power type. They showed in particular that there is a uniformly convex function f bounded above by ‖ ⋅ ‖ if and only if there is an equivalent norm on X with power type 2. Recall that a real-valued convex function f defined on a nonempty convex subset D of X is said to be uniformly convex provided that for every ε > 0 there is δ > 0 such that f(x) + f(y) − 2f((x + y)/2) ≥ δ whenever x, y ∈ D with ‖x − y‖ ≥ ε. An application of uniformly convex functions can be found in [13]. In the present note we are interested in characterizing reflexivity by convex functions. As a result, we prove that a Banach space being reflexive is equivalent to the fact that there exists a continuous function with some kind of convexity on some nonempty open convex subset of the space. We review recent works in this direction. Odell and Schlumprecht’s renorming theorem [14] shows that a separable Banach space X is reflexive if and only if there is an equivalent 2R norm ‖ ⋅ ‖ on X; that is, if a bounded sequence


Introduction
Like differentiability, convexity of functions has proven to be a very useful tool to characterize various classes of Banach spaces.We have also known that good progress has been made in this direction.See, for instance, [1][2][3][4][5][6][7][8][9][10].In particular, Zȃlinescu [10] first proved that a Banach space is reflexive provided that there exists a continuous uniformly convex function on some nonempty open convex subset of the space.Later, Cheng et al. [11] proved that, in fact, such a Banach space is super reflexive, and vice versa (recall that a Banach space is super reflexive if and only if it admits an equivalent uniformly convex norm [12]).Recently, this result has also been obtained independently by Borwein et al. (see [2,Theorem 2.4]).In particular, in [2] they mainly investigated the relationship between the existence of uniformly convex functions  :  → R bounded above by some power type of the norm and the existence of an equivalent norm with a certain power type.They showed in particular that there is a uniformly convex function  bounded above by ‖ ⋅ ‖ 2 if and only if there is an equivalent norm on  with power type 2. Recall that a real-valued convex function  defined on a nonempty convex subset  of  is said to be uniformly convex provided that for every  > 0 there is  > 0 such that () + () − 2(( + )/2) ≥  whenever ,  ∈  with ‖ − ‖ ≥ .An application of uniformly convex functions can be found in [13].
In the present note we are interested in characterizing reflexivity by convex functions.As a result, we prove that a Banach space being reflexive is equivalent to the fact that there exists a continuous function with some kind of convexity on some nonempty open convex subset of the space.
We review recent works in this direction.Odell and Schlumprecht's renorming theorem [14] shows that a sepa- then (  ) is norm convergent in .More recently, Hájek and Johanis [15], through introducing a new convexity property of Day's norm on  0 (Γ), showed the following renorming characterization of general reflexive spaces.A sufficient and necessary condition for a Banach space  to be reflexive is that it admits an equivalent 2 norm; that is, if a bounded sequence (  ) ⊂  satisfies (1), then (  ) is weakly convergent in .The localized versions of the two renorming theorems have been considered in [16].
The purpose of the present note is to provide a new characterization of reflexive Banach spaces by convex functions.More precisely, the following is our main result.

Theorem 1. (i) A Banach space 𝑋 is reflexive if (and only if) there exists a continuous 𝑤2𝑅 convex function on some nonempty open convex set 𝐷 of 𝑋.
(ii) In particular, if  is separable then  is reflexive if (and only if) there exists a continuous 2 convex function on some nonempty open convex set  of .
Our notation and terminology for Banach spaces are standard, as may be found for example in [17,18].All Banach spaces throughout the paper are supposed to be real.The letter  will always denote a Banach space and  * its dual.

Proof of the Main Theorem
In order to complete the proof of our main theorem, we need to introduce some notation and make some preparatory remarks.
The following notion is a natural generalization of 2 (resp., 2) norm.Definition 2. Let  be a nonempty convex set of a Banach space .A real-valued convex function  defined on  is said to be 2 (2, resp.); if a bounded sequence (  ) ⊂  satisfies then (  ) is norm convergent (resp., weakly convergent) in .
Assume that  ia a real-valued convex function defined on a nonempty convex subset  of .Recall that the subdifferential of  at  ∈  is the set We have already known (see, e.g., [8]) that if the convex function  is continuous at  ∈  then () is a nonempty  * -compact convex subset of  * .
Proof.Indeed, under the assumption of the lemma we claim that  is Lipschitz on [ 0 , +∞) for every  0 > 0.

Lemma 4.
Let  be a bounded closed convex subset of a Banach space .Assume that there is a continuous 2 convex function on .Then  is a weakly compact set.
Proof.Without loss of generality, we can assume that 0 ∈ .
Proof of Theorem 1.The "if" part of (i) is as follows: we assume without loss of generality that 0 ∈  and that (0, ) = { ∈  : ‖‖ ≤ } ⊂ .By Lemma 4, the set (0, ) is weakly compact, and hence  is reflexive.This also immediately implies that the "if" part of (ii) is valid.
The "only if" parts of (i) and of (ii) are due to the aforementioned renorming theorems of Hájek and Johanis [15] and Odell and Schlumprecht [14], respectively.Remark 5.It should be mentioned that, especially recently, several characterizations of reflexivity were also obtained using purely metric properties of Banach spaces (but not the linear structure).See, for example, [21][22][23].The interested reader may refer to those papers for further information.