Bornological LB-spaces and idempotent adjunctions

TL;DR

This paper compares two recent notions of bornological LB-spaces, showing they are distinct yet related, through an analysis of the bornologification and topologification functors and their idempotent adjunctions.

Contribution

It clarifies the relationship between two definitions of bornological LB-spaces and studies the associated idempotent adjunctions in detail.

Findings

The two notions of bornological LB-spaces are not equivalent.

The two concepts are closely related despite their differences.

Analysis of bornologification and topologification functors reveals their idempotent adjunctions.

Abstract

The notion of an LB-space was introduced by Grothendieck in his 1953 th\`{e}se, referring to a countable colimit of Banach spaces taken within the category of locally convex topological vector spaces, and refining prior work done by Dieudonn\'{e}, Schwartz and K\"othe. Recently, two different notions of `bornological LB-spaces' emerged: one, given by Stempfhuber, refers to countable colimits of Banach spaces as well, but now taken in the category of bornological vector spaces. The other one, given by Bambozzi, Ben-Bassat and Kremnizer, refers to bornologifications of regular LB-spaces, i.e., of LB-spaces in the Grothendieck sense having the additional property that every bounded subset of the colimit is contained and bounded in one of its Banach steps. In this note, we show that the two notions are distinct, but nevertheless closely related. This involves, in particular, an intimate study of the idempotent adjunction of the bornologification and topologification functors.