  We discuss possible definitions of categories of vector spaces over a fixed
field $\mathbf{k}$ 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. We call these \emph{categories of reasonable
strong vector spaces} (c.r.s.v.s.). We show that, in a precise sense, the more
general possible definition for a reasonable strong vector space is that of a
small $\mathbf{k}$-additive endofunctor of $\mathrm{Vect}_\mathbf{k}$ such that
for every cardinal $\lambda$ there are no non-zero natural transformations from
$\mathrm{Vect}(\mathbf{k},-)^\lambda /\mathrm{Vect}(\mathbf{k}, -)^{\oplus
\lambda}$ to $X$. We denote the category such functors $\Sigma\mathrm{Vect}$.
We compare $\Sigma\mathrm{Vect}$ to the category of ``naive" strong vector
spaces $B\Sigma\mathrm{Vect}$ constructed in a specific way and to the
c.r.s.v.s. $K\mathrm{TVect}_s$ of separated linearly topologized spaces that
are generated by linearly compact spaces. We study the number of iterations of
a natural approximate reflector on
$\mathrm{Ind}\text{-}(\mathrm{Vect}^{\text{op}})$ needed to construct the
orthogonal reflector onto $\Sigma\mathrm{Vect}$ because it relates to the
problem of constructing the smallest subspace of an $X \in \Sigma\mathrm{Vect}$
closed under taking infinite linear combinations containing a given linear
subspace of $H$ of $X$. We give a complete answer in the case $X$ is
$\aleph_1$-ary. Finally we show how the natural monoidal closed structure on
$\mathrm{Ind}\text{-}(\mathrm{Vect}^{\text{op}})$ induces one on
$\Sigma\mathrm{Vect}$ and apply this to define an infinite-sum-sensitive notion
of K\"ahler differentials for generalized power series. Most of the technical
results apply to a more general class of reflective subcategories of
$\mathrm{Ind}\text{-}(\mathrm{Vect}^{\text{op}})$.
