Completeness of regular LB-spaces

Establish whether every regular LB-space is complete, i.e., determine if the inductive limit of a countable sequence of Banach spaces that is regular in the sense of bounded sets being contained and bounded in some step is necessarily complete.

Background

Grothendieck’s classical completeness problem asks whether regular inductive limits of Fréchet/Banach spaces are complete. This paper focuses on the LB-case (inductive limits of Banach spaces) and studies homological formulations, but the core analytic question remains unresolved. The authors emphasize the longevity of this problem and its roots in early functional analysis.

References

We consider the long-standing question of whether every regular LB-space is complete. The latter is open since the 1950s and is based on Grothendieck's early work in functional analysis.

— A homological approach to (Grothendieck's) completeness problem for regular LB-spaces  (2512.13161 - Wegner, 15 Dec 2025) in Abstract; Section 1 (Introduction)