On the relation between distances and seminorms on Fr\'echet spaces, with application to isometries

TL;DR

This paper investigates the structure of linear isometries on Fréchet spaces with metrics defined by seminorms, providing conditions under which such isometries preserve individual seminorms and extending classical theorems to broader contexts.

Contribution

It establishes growth conditions on the metric-defining function ensuring isometries preserve seminorms, extending the Banach--Stone theorem to more general spaces.

Findings

Conditions for isometries to preserve seminorms in Fréchet spaces

Characterizations of isometries on spaces of holomorphic functions

Extension of the Banach--Stone theorem to non-surjective cases

Abstract

A study is made of linear isometries on Fr\'echet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach--Stone theorem to surjective and nonsurjective cases.