On the axiomatization of convex subsets of Banach spaces (original) (raw)

On 𝑃- and 𝑝-Convexity of Banach Spaces

Abstract and Applied Analysis, 2010

We show that everyU-space and every Banach spaceXsatisfyingδX(1)>0areP(3)-convex, and we study the nonuniform version ofP-convexity, which we callp-convexity.

New concepts of convexity in Banach spaces

2014

In this paper, are provided a new set in Banach spaces. Using McShane integral, new concepts of convexity are introduced in Banach spaces. These results are generalized concepts that Jian-Yong Wang and Yu-Mei Ma were expressed. Convex sets and extreme points have a very important role in optimal control problems.By way of McShane integral of vector-valued functions, the McShane integral convexity of sets and the concept of McShane integral extreme points of sets are introduced in banach spaces.

On Functionally Convex Sets and Functionally Closed Sets in Real Banach Spaces

Pure and Applied Mathematics, 2018

We have introduced two new notions of convexity and closedness in functional analysis. Let X be a real normed space, then C(⊆ X) is functionally convex (briefly, F-convex), if T (C) ⊆ R is convex for all bounded linear transformations T ∈ B(X, R); and K(⊆ X) is functionally closed (briefly, F-closed), if T (K) ⊆ R is closed for all bounded linear transformations T ∈ B(X, R). By using these new notions, the Alaoglu-Bourbaki-Eberlein-Šmuljan theorem has been generalized. Moreover, we show that X is reflexive if and only if the closed unit ball of X is F-closed. James showed that for every closed convex subset C of a Banach space X, C is weakly compact if and only if every f ∈ X * attains its supremum over C at some point of C. Now, we show that if A is an F-convex subset of a Banach space X, then A is bounded and F-closed if and only if every element of X * attains its supremum over A at some point of A.

Some results on functionally convex sets in real Banach spaces

2016

We use of two notions functionally convex (briefly, F-convex) and functionally closed (briefly, F-closed) in functional analysis and obtain more results. We show that if {A α } α∈I is a family Fconvex subsets with non empty intersection of a Banach space X, then ∪ α∈I A α is F-convex. Moreover, we introduce new definition of notion F-convexiy.

The Knaster–Kuratowski–Mazurkiewicz theorem and abstract convexities

Journal of Mathematical Analysis and Applications, 2008

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called "intersection" theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from R n to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.

Functionally closed sets and functionally convex sets in real Banach spaces

International Journal of Nonlinear Analysis and Applications, 2016

‎Let XXX be a real normed space, then C(subseteqX)C(subseteq X)C(subseteqX) is functionally convex (briefly, FFF-convex), if T(C)subseteqBbbRT(C)subseteq Bbb R T(C)subseteqBbbR is convex for all bounded linear transformations TinB(X,R)Tin B(X,R)TinB(X,R); and K(subseteqX)K(subseteq X)K(subseteqX) is functionally closed (briefly, FFF-closed), if T(K)subseteqBbbRT(K)subseteq Bbb R T(K)subseteqBbbR is closed for all bounded linear transformations TinB(X,R)Tin B(X,R)TinB(X,R). We improve the Krein-Milman theorem on finite dimensional spaces. We partially prove the Chebyshev 60 years old open problem. Finally, we introduce the notion of functionally convex functions. The function fff on XXX is functionally convex (briefly, FFF-convex) if epi fff is a FFF-convex subset of XtimesmathbbRXtimes mathbb{R}XtimesmathbbR. We show that every function f:(a,b)longrightarrowmathbbRf : (a,b)longrightarrow mathbb{R}f:(a,b)longrightarrowmathbbR which has no vertical asymptote is FFF-convex.

Uniformly convex Banach spaces are reflexive—constructively

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive. Our aim in this note is to present a fully constructive 1 analysis of the Milman-Pettis theorem [11, 12, 9, 13]: a uniformly convex Banach space is reflexive. First, though, we need to outline the constructive context of the statement and proof of this theorem, and to clarify the terms we will use. The context of our work is that of a quasinormed space: that is, a linear space X equipped with a family (i) i∈I of seminorms on X such that the subset {{x i : i ∈ I} of R is bounded; that family is called the quasinorm on X. For all x, x in the quasinormed space X we define the inequality and equality relations by: • (x = x) ≡ ∃ i∈I (x − x i > 0); and • (x = x) ≡ ∀ i∈I (x − x i = 0) .

A New Class of Banach Spaces and Its Relation with Some Geometric Properties of Banach Spaces

Abstract and Applied Analysis, 2012

By introducing the concept ofL-limited sets and thenL-limited Banach spaces, we obtain some characterizations of it with respect to some well-known geometric properties of Banach spaces, such as Grothendieck property, Gelfand-Phillips property, and reciprocal Dunford-Pettis property. Some complementability of operators on such Banach spaces are also investigated.