Mazur-Ulam’s classical theorem states that any isometric onto map between normed spaces is linear. This result has been generalized by T. Figiel [F] who showed that ifΦ is an isometric embedding from a Banach space X to a Banach space Y such that Φ(0) = 0 and vect [φ(X )] = Y , there exists a linear quotient map Q such that ‖Q‖ = 1 and Q ◦Φ= I dX . The third chapter of this short story is [GK] ...

The Banach-Stone theorem states that any surjective, linear mapping T between spaces of continuous functions that satisfies ‖T (f)− T (g)‖ = ‖f − g‖, where ‖ · ‖ denotes the uniform norm, is a weighted composition operator. We study a multiplicative analogue, and demonstrate that a surjective mapping T , not necessarily linear, between algebras of continuous functions with ‖T (f)T (g)‖ = ‖fg‖ m...

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...

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...