This category has vector spaces over as objects, and -linear maps between these as morphisms.
Alternatively, one sometimes defines “vector space” as a two-sorted notion; taking the field as one of the sorts and a module over as the other. More generally, the notion of “module” can also be considered as two-sorted, involving a ring and a module over that ring.
This is occasionally convenient; for example, one may define the notion of topological vector space or topological module as an internalization in of the multisorted notion. This procedure is entirely straightforward for topological modules, as the notion of module can be given by a two-sorted Lawvere theory , whence a topological module (for instance) is just a product-preserving functor . One may then define a topological vector space as a topological module whose underlying (discretized) ring sort is a field.
vitual vector space?
vector space, dual vector space,