Year: 2025
Type: preprint
Abstract: We consider the long-standing question of whether every regular LB-space is complete. This problem has been open since the 1950s and originates in Grothendieck's early work in functional analysis. Rather than seeking a direct proof or counterexample, our approach is to study weak versions of the problem using homological methods. We consider the categories of complete and, respectively, regular LB-spaces, establish that their derived categories are well-defined with respect to several exact structures, and show that there are canonical triangle functors between them. If one of these functors were not an equivalence, this would provide a negative answer to Grothendieck's question. In contrast, we prove that one of them is an equivalence. This may be interpreted as evidence in favor of an affirmative answer to the original problem, and it shows in particular that the two classes of spaces share the same homological algebra, even if they were to differ.
Source: arXiv (Cornell University)
Author: Wegner, Sven-Ake
Institutions: Universität Hamburg, HAW Hamburg
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4417455703
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