# Extension of differentiable local mappings on linear topological spaces

## TL;DR

This paper explores a new method for extending local maps in linear topological spaces that lack smooth bump functions, generalizing previous approaches from Banach spaces to broader contexts.

## Contribution

It introduces a generalized approach for extending local maps in arbitrary linear topological spaces, expanding beyond Banach spaces.

## Key findings

- The new method is applicable to a wider class of topological spaces.
- It provides a framework for extending local maps without smooth bump functions.
- Potential applications in infinite-dimensional analysis are discussed.

## Abstract

Usually, for extension of local maps, one uses multiplication by so called bump functions. However, majority of infinite-dimensional linear topological spaces do not have smooth bump functions. Therefore, in \cite{BR} we suggested a new approach for Banach spaces, based on the composition with locally identical maps. In the present work we discuss a possibility of generalization of this method for arbitrary spaces and applications of this theory.