A generalization of Lipschitz mappings

TL;DR

This paper introduces extensively bounded mappings, a generalization of Lipschitz mappings, along with a new metric and norm, and explores their properties, linearizations, and associated operator ideals.

Contribution

It extends the concept of Lipschitz mappings using modulus of continuity, defines a new metric and norm, and develops the theory of extensively bounded operator ideals.

Findings

Introduces extensively bounded mappings as a generalization of Lipschitz mappings.

Defines a new metric and norm for these mappings, with properties studied.

Develops the theory of extensively bounded operator ideals, including finite rank and compact mappings.

Abstract

Using the notion of modulus of continuity at a point of a mapping between metric spaces, we introduce the notion of extensively bounded mappings generalizing that of Lipschitz mappings. We also introduce a metric on it which becomes a norm if the codomain is a normed linear space. We study its basic properties. We also discuss a linearization of an extensively bounded mapping into a bounded linear mapping. As an application, we introduce the notion of extensively bounded operator ideals. We also discuss extensively bounded finite rank and extensively bounded compact mappings and their corresponding operator ideals.