I got the feeling that some of the "usual categories" in functional analysis could be homological (though, excuse my ignorance, I don't know anything about functional analysis, yet). E.g. in "Lectures and Exercises on Functional Analysis" by Helemskii the author considers exact sequences of Banach spaces.

To be more specific I'm asking about some of the, apparently, more popular ones. Are any of the following homological?

  • Topological vector spaces with continuous linear maps
  • Normed vector spaces with bounded operators
  • Banach spaces with bounded operators
  • Hilbert spaces with bounded operators
  • Banach algebras with bounded morphisms

(any others come to mind?)

Partial answers are fine! I merely don't think it is a good idea to ask 5 separate questions with potentially overlapping answers.

Background info:

A category with finite limits and finite colimits is homological if and only if:

  • regular epis (coequalizers) are stable under pullbacks
  • there is a zero object
  • the short five lemma holds

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