Homological categories in functional analysis

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.

I suspect topological vector spaces and normed spaces are also homological but have not worked out the details. They would require getting some messier work since they do not have the open mapping theorem (i.e., to show a morphism is an isomorphism, it's not enough to just show it's a bijection).