The category of Silva spaces is not integral

We establish that the category of Silva spaces, aka $\mathrm{LS}$-spaces, formed by countable inductive limits of Banach spaces with compact linking maps as objects and linear and continuous maps as morphisms, is not an integral category. The result carries over to the category of $\mathrm{PLS}$-spaces, i.e., countable projective limits of $\mathrm{LS}$-spaces—which contains prominent spaces of analysis such as the space of distributions and the space of real analytic functions. As a consequence, we obtain that both categories neither have enough projective nor enough injective objects. All results hold true when ‘compact’ is replaced by ‘weakly compact’ or ‘nuclear’. This leads to the categories of $\mathrm{PLS}$-, $\mathrm{PLS_w}$- and $\mathrm{PLN}$-spaces, which are examples of ‘inflation exact categories with admissible cokernels’ as recently introduced by Henrard, Kvamme, van Roosmalen and the second-named author.

Keywords

$\mathrm{LS}$-space, $\mathrm{PLS}$-space, integral category, injective or projective object

2010 Mathematics Subject Classification

18Gxx, 46A45, 46M10, 18E05, 46A13

Full Text (PDF format)

Received 9 February 2022

Received revised 24 April 2022

Accepted 2 May 2022

Published 2 May 2023