# Extension of local smooth maps of Banach spaces

## TL;DR

This paper introduces K-maps as a substitute for bump functions in infinite-dimensional Banach spaces, enabling the extension of smooth local maps in spaces lacking smooth bump functions, and proves a Borel lemma for such spaces.

## Contribution

It proposes K-maps as a novel tool to extend local maps in Banach spaces without smooth bump functions and establishes a Borel lemma for these spaces.

## Key findings

- K-maps effectively replace bump functions in certain Banach spaces.
- Extension of smooth local maps is possible using K-maps in non-smooth spaces.
- The Borel lemma is proven for spaces with K-maps.

## Abstract

It is known that smooth bump functions are absent in the majority of infinite-dimensional Banach spaces. This is an obstacle in the development of local analysis, in particular in the questions of extending local maps onto the whole space. We suggest an approach that substitutes bump functions with special maps, which we call K-maps. It allows us to extend smooth local maps from non-smooth spaces, such as $C^q[0,1], q=0,1,...$. We also prove the Borel lemma for spaces possessing K-maps.