On Vector Spaces with Formal Infinite Sums

Abstract: I discuss possible definitions of categories of vector spaces enriched with a notion of formal infinite linear combination in the likes of the formal infinite linear combinations one has in the context of generalized power series, I call these \emph{reasonable categories of strong vector spaces} (r.c.s.v.s.). I show that, in a precise sense, the more general possible definition for a strong vector space is that of a small $\mathrm{Vect}$-enriched endofunctor of $\mathrm{Vect}$ that is right orthogonal for every cardinal $\lambda$, to the cokernel of the canonical inclusion of the $\lambda$-th copower in the $\lambda$-th power of the identity functor: these form the objects for a universal r.c.s.v.s. I call $\Sigma\mathrm{Vect}$. I show this is equivalent to the category of \emph{ultrafinite summability spaces} defined independently in arXiv:2403.05827. I relate this category to what could be understood to be the obvious category of strong vector spaces $B\Sigma\mathrm{Vect}$ and to the r.c.s.v.s. $K\mathrm{TVect}_s$ of separated linearly topologized spaces that are generated by linearly compact spaces. I analyze the monoidal closed structures on various r.s.v.s. induced by the natural one on $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$. In particular with respect to the problem of closure under the tensor product of $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$. Most of the technical results apply to a more general class of orthogonal subcategories of $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$ and we work with that generality as it's cost-free.