A new proof for the bornologicity of the space of slowly increasing functions

TL;DR

This paper provides a novel proof that the space of slowly increasing functions is bornological, employing homological methods and the derived projective limit functor, avoiding reliance on sequence space isomorphisms.

Contribution

It introduces the first proof of bornologicity for these function spaces that does not depend on their isomorphism to sequence spaces, using homological techniques.

Findings

Established the bornologicity of the space of slowly increasing functions without sequence space isomorphisms

Applied homological methods and derived functors in functional analysis

Provided a new perspective on the structure of these function spaces

Abstract

A. Grothendieck proved at the end of his thesis that the space of slowly increasing functions and the space of rapidly decreasing distributions are bornological. Grothendieck's proof relies on the isomorphy of these spaces to a sequence space and we present the first proof that does not utilize this fact by using homological methods and, in particular, the derived projective limit functor.