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references on bornological vector spaces
Christian Houzel. Espaces analytiques relatifs et throrrme de finitude. Math. Ann. v.205 (1973), p.13-54.
Ralf Meyer. Analytic cyclic cohomology (1999). (arXiv:math/9906205).
Ralf Meyer. Embeddings of derived categories of bornological modules (2004). (arXiv:math/0410596).
moving the references on bornological vector spaces to the entry bornological topological vector space
This is probably a bit off-topic but the only sensible thing I can say is from the point of view of differential categories: From the paper A convenient differential category, the category of “Mackey-complete, separated, topological convex bornological vector spaces and bornological linear maps is a differential category”. They also say that this category is symmetric monoidal closed, complete and cocomplete. I believe (this is not something which is published, and I have not yet worked carefully on this. It is not the same that the better known fact that $Vect^{op}$ is a differential category.) that $Vect$ is also a differential category but with a map $!A \rightarrow B$ being something called a “polynomial law”. $Vect$ is also a symmetric monoidal closed, complete and cocomplete category. So as far as being a differential category, monoidal closed, complete and cocomplete are concerned, this category of bornological spaces have the same categorical properties than $Vect$ but it allows to treat more differentiable functions.
This may be worthwhile recording at bornological vector space!
I think that generically any category of structured vector spaces will tend to have less good categorical properties than plain $Vect$. The fact worth recording is if such categories are about as well behaved as plain $Vect$, not the other way around.
Did not realize we already had an entry for this. I recorded this references because of their use in and Prop 3.6 in
Yes, these kinds of propositions are noteworthy not because $Vect$ would not have these properties, but because it is non-trivial for topological vector spaces to still have such properties.
(In this vein, currently the whole field of “condensed mathematics” is being developed to make algebra with topological structure retain the good category-theoretic properties of plain algebra.)
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