We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach...

The numbers of 2-connected and 2-edge-connected Steinhaus graphs with n vertices are determined for n¿ 6. The possible generating strings of 2-connected and 2-edge-connected Steinhaus graphs are classi0ed. c © 2002 Elsevier Science B.V. All rights reserved.

This paper deals with some new generalizations of Farkas' theorem for a class of set-valued mappings with arbitrary convex cones in infinite-dimensional Banach spaces. A modified Farkas' theorem with no closedness assumption is given. The generalized Gale alternative theorem in nonlinear programming is derived as an easy consequence. The results are applied to constrained controllability theory...

In this paper, we present a new extension of Banach-Caristi type theorem for multivalued mappings. We show that our result is not consequence version Banach contraction principle due to Nadler. provide an application the solution functional equations.

Introduction. The question of ergodicity of a semigroup of bounded linear operators on a Banach space has been reduced, by Alaoglu and Birkhoff [l],1 Day [2, 3], and Eberlein [4], to the study firstly of the ergodicity of the semigroup itself and secondly, of the ergodicity of each element of the Banach space with respect to this ergodic semigroup. In the case of a bounded and commutative semig...

This lemma was then used the following year as a crucial step in the proof of the well-known Bishop-Phelps theorem [1] that every Banach space is subreflexive; in other words, every functional on a Banach space E can be approximated by a norm-attaining functional on the same space. The original proof of this lemma uses the Hahn-Banach theorem and is therefore fairly abstract. In this note, we p...

A well known application of Ramsey’s Theorem to Banach Space Theory is the notion of a spreading model (ẽi) of a normalized basic sequence (xi) in a Banach space X. We show how to generalize the construction to define a new creature (ei), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normaliz...

in this paper, we first introduce the notion of $c$-affine functions for $c> 0$.then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. moreover, a hyers–-ulam stability result for strongly convex functions is shown.