It is well-known that the notion of a rigid monoidal category does not interact well with infinite-dimensional spaces. To be more precise, the category of all vector spaces is not rigid, see for example question 1 or question 2.
Are there any related and/or weaker notions of "rigid monoidal category" which would allow consideration of infinite dimensional vector spaces (for example) in some sense?
I am not a category theorist, but have a functional analysis background. I would be very happy to have an answer which used extra structure on the spaces (e.g. topology? a (complete) inner-product?)
Update: For example, consider the category of Banach spaces with bounded linear maps (or contractions, if you prefer) and the projective tensor product to obtain a symmetric monoidal category. This nearly satisfies the conditions to be a star-autonomous category with the usual Banach space dual, as it is true that $$ \mathcal B(E \widehat\otimes F, G^*) \cong \mathcal B(E, (F\widehat\otimes G)^*). $$ However, the map $*$ is only a faithful functor, not full, as $\mathcal B(E,F) \rightarrow \mathcal B(F^*,E^*), T\mapsto T^*$ is not surjective, in general. I think one can get further by using the Chu construction.
However, somehow this example feels wrong to me. I think the reason why is that I'd like to:
Capture both the evaluation ($X^*\otimes X\rightarrow 1$) and coevaluation ($1\rightarrow X^*\otimes X$) maps, in some sense.