# Homological categories in functional analysis

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
• Note that "regular epis are stable under pullbacks" does not mean the same thiing as "coequalizers are stable under pullbacks"--see math.stackexchange.com/questions/1494439/…. Jul 2, 2019 at 22:15

Banach spaces, Hilbert spaces, and (non-unital) Banach algebras are all homological. In these categories regular epis are the same as surjective morphisms (by the open mapping theorem, if $$f:X\to Y$$ is a bounded linear surjection of Banach spaces then the induced map $$X/\ker f\to Y$$ is a topological isomorphism), and so it follows easily that they are stable under pullbacks (pullbacks can be computed as the pullback of sets, with the norm induced by the inclusion into the product). To verify the short five lemma, note that the short five lemma for plain vector spaces implies the middle map will be a vector space isomorphism, and then the open mapping theorem says the inverse is bounded.