The idea behind this article is to provide a unified and relatively nontechnical framework for treating the main existence theorems for continuous linear functionals in functional analysis, convex analysis, Lagrange multiplier theory, and minimax theory. While many of the results in this article are already known, our approach is new, and gives a large number of results with considerable econom...

We obtain an analogue of Wigner’s classical theorem on symmetries for Banach spaces. The proof is based on a result from the theory of linear preservers. Moreover, we present two other Wigner-type results for finite dimensional linear spaces over general fields. Wigner’s theorem on symmetry transformations (sometimes called unitary-antiunitary theorem) plays fundamental role in quantum mechanic...

0. Introduction 2 The impact of logic in Banach space theory 2 The case of model theory 2 Model theory for structures of functional analysis 3 Two famous applications 4 A note on the exposition 4 1. Preliminaries: Banach Space Models 5 Banach space structures and Banach space ultrapowers 5 Positive bounded formulas 7 Approximate satisfaction 8 (1 + )-isomorphism and (1 + )-equivalence of struct...

In this paper, we partially solve an open problem, due to J. C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer n, that are Steinhaus triangles containing all the elements of Z/nZ with the same multiplicity. For every odd number n, we build an orbit in Z/nZ, by the linear cellular automaton generating the Pascal triangle modulo n, which contains infi...

The two main results are: A. If a Banach space X is complementably universal for all subspaces of c0 which have the bounded approximation property, then X∗ is non separable (and hence X does not embed into c0), B. There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X. Theorem B solves a problem that dates from the 1970s.

The Hahn-Banach theorem in its simplest form asserts that a bounded linear functional defined on a subspace of a Banach space can be extended to a linear functional defined everywhere, without increasing its norm. There is an order-theoretic version of this extension theorem (Theorem 0.1 below) that is often more useful in context. The purpose of these lecture notes is to discuss the noncommuta...

The Central Sets Theorem is a powerful theorem, one of whose consequences is that any central set in N contains solutions to any partition regular system of homogeneous linear equations. Since at least one set in any finite partition of N must be central, any of the consequences of the Central Sets Theorem must be valid for any partition of N. It is a result of Beiglböck, Bergelson, Downarowicz...

We first introduce a modified proximal point algorithm for maximal monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of maximal monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality probl...

The Bishop-Phelps theorem [1] states that for any Banach space X, the set of norm attaining linear bounded functionals is dense in X ′, the dual space of X. Since then, the study of norm attaining functions has attracted the attention of many authors. Lindenstrauss showed in [2] that there is no Bishop-Phelps theorem for linear bounded operators. Nevertheless, he proved that the set of bounded ...