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Banach-Stone type theorems on uniformly continuous and lipschitz continuous pseudometrics

Katsuhisa Koshino

Abstract

In this paper, we shall establish Banach-Stone type theorems on spaces of uniformly continuous and lipschitz continuous pseudometrics.

Banach-Stone type theorems on uniformly continuous and lipschitz continuous pseudometrics

Abstract

In this paper, we shall establish Banach-Stone type theorems on spaces of uniformly continuous and lipschitz continuous pseudometrics.
Paper Structure (6 sections, 13 theorems, 88 equations)

This paper contains 6 sections, 13 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Uniformities and pseudometrics
  3. The peaking function argument
  4. Proof of Theorems A, B and C
  5. Spaces of admissible metrics
  6. Problem

Key Result

Theorem 1.1

Suppose that $X = (X,d_X)$ and $Y = (Y,d_Y)$ are complete metric spaces. The following are equivalent: In this case, for each isometry $T : \operatorname{UC}(X,d_X) \to \operatorname{UC}(Y,d_Y)$, there exists a uniform homeomorphism $\phi : Y \to X$ and $\alpha \in \operatorname{UC}(Y,d_Y)$ with $\alpha(Y) \subset \{1,-1\}$ such that for any $f \in \operatorname{UC}(X,d_X)$ and for any $y \in Y$,

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 14 more