Characterization of Reflexivity by Convex Functions
A new convexity property of convex functions is introduced. This property provides, in particular, a characterization of the class of reflexive Banach spaces.
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  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.  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 ). Recently, this result has also been obtained independently by Borwein et al. (see [2, Theorem ]). In particular, in  they mainly investigated the relationship between the existence of uniformly convex functions 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 if and only if there is an equivalent norm on with power type . Recall that a real-valued convex function defined on a nonempty convex subset of is said to be uniformly convex provided that for every there is such that whenever with . An application of uniformly convex functions can be found in .
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  shows that a separable Banach space is reflexive if and only if there is an equivalent norm on ; that is, if a bounded sequence satisfies then is norm convergent in More recently, Hájek and Johanis , through introducing a new convexity property of Day’s norm on , 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 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 .
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 is reflexive if (and only if) there exists a continuous convex function on some nonempty open convex set of .
(ii) In particular, if is separable then is reflexive if (and only if) there exists a continuous 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.
2. 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 (resp., ) norm.
Definition 2. Let be a nonempty convex set of a Banach space A real-valued convex function defined on is said to be (, resp.); if a bounded sequence satisfiesthen 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 setWe have already known (see, e.g., ) that if the convex function is continuous at then is a nonempty -compact convex subset of
Lemma 3. Let be a bounded closed convex set in with . Given an let be defined byHere is a (real-valued) nonnegative continuous convex function on with . Then is continuous on .
Proof. Indeed, under the assumption of the lemma we claim that is Lipschitz on for every . To see this, fix a and let such that . Put . By the definition of one can find an element with such thatWe next distinguish two cases. First, if , it follows from (3) and (4) that Second, if , then we can find a such that and Indeed, let , for all . The continuity of on ensures that is also continuous on From and , it follows that there exists a such that . That is, , and the convexity of gives . Thuswhere
In short we haveThe convexity of , combined with together, implies that . ThereforePutting this and (8) together, we havewhere . This completes our claim. Therefore, is continuous on
Lemma 4. Let be a bounded closed convex subset of a Banach space Assume that there is a continuous convex function on . Then is a weakly compact set.
Proof. Without loss of generality, we can assume that . The continuity of convex function on implies that , the subdifferential of at , is not empty. Take and let be defined byThen is a nonnegative continuous convex function on with . Clearly, is also convex on . By the James characterization for weakly compact sets [19, 20], it suffices to prove that every linear functional attains its maximum on For any given , we can assume that (otherwise we have ). Let be defined byBy Lemma 3, is continuous on . SettingIn the following, through discussing two cases of , that is, and , we want to deduce that can always obtain its maximum on .
Case 1. If , let such that for all . Choose a sequence such that and such thatHence . The convexity of immediately implies thatThe property of on gives thatfor some This, combined with (14), means that
Case 2. If . Choose a sequence satisfying . Then for all . The continuity of on gives that , and hence Thus . Let with such that . Since is bounded, there exists a subsequence, still denoted by , so that it is convergent. Obviously , and yet,Therefore is weakly convergent in , say, , for some . So we have . This completes the proof.
Now, we can prove the main result of this note.
Proof of Theorem 1. The “if” part of (i) is as follows: we assume without loss of generality that and that By Lemma 4, the set 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  and Odell and Schlumprecht , 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, [21–23]. The interested reader may refer to those papers for further information.
The authors declare that they have no competing interests.
Zhenghua Luo was supported partially by the Natural Science Foundation of China, Grant no. 11201160, and the Natural Science Foundation of Fujian Province, Grant no. 2012J05006. Qingjin Cheng was supported in part by the Natural Science Foundation of China, Grant nos. 11471271 and 11371296.
J. M. Borwein and S. A. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, vol. 3 of CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, Springer, New York, NY, USA, 2nd edition, 2006.
R. Deville, G. Godefroy, and V. Zizler, Smoothness and Renormings in Banach Spaces, vol. 64 of Pitman Monographs and Surveys in Pure and Applied, Mathematics, Longman Scientific & Technical, John Wiley & Sons, New York, NY, USA, 1993.
R. R. Phleps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2nd edition, 1993.
P. Enflo, “Banach spaces which can be given an equivalent uniformly convex norm,” Israel Journal of Mathematics, vol. 13, pp. 281–288, 1972.View at: Google Scholar
S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 313–318, Dekker, New York, NY, USA, 1996.View at: Google Scholar | MathSciNet
J. Lindenstrauss and L. Trafiri, Classical Banach Spaces. I. Sequence Spaces, vol. 92 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, Germany, 1977.
J. Lindenstrauss and L. Trafiri, Classical Banach Spaces. II. Function Spaces, vol. 97 of Results in Mathematics and Related Areas, Springer, Berlin, Germany, 1979.