Approximation of continuous functions on Fr\'echet spaces

TL;DR

This paper investigates how continuous functions on countable normed Fréchet spaces can be approximated by analytic functions and provides criteria for extending continuous functions from dense subspaces.

Contribution

It introduces methods for approximating continuous functions with analytic functions and establishes a criterion for extending functions from dense subspaces in topological spaces.

Findings

Approximation of continuous functions by analytic and *-analytic functions in Fréchet spaces

A criterion for extending continuous functions from dense subspaces

Insights into the structure of function extension in topological spaces

Abstract

We consider approximations of a continuous function on a countable normed Fr\'{e}chet space by analytic and $*$-analytic. Also we found a criterium of the existence of an extension of a continuous function from a dense subspace of a topological space onto the space.