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For kk a field, a vector space over kk is module over the ring kk. Sometimes a vector space over kk is called a kk-linear space. (Compare ‘kk-linear map’.)

The category of vector spaces is typically denoted Vect, or Vect kVect_k if we wish to make the field kk (the ground field) explicit. So

Vect kkMod. Vect_k \coloneqq k Mod \,.

This category has vector spaces over kk as objects, and kk-linear maps between these as morphisms.

Multisorted notion

Alternatively, one sometimes defines “vector space” as a two-sorted notion; taking the field kk as one of the sorts and a module over kk 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 TopTop 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 TT, whence a topological module (for instance) is just a product-preserving functor TTopT \to Top. One may then define a topological vector space as a topological module whose underlying (discretized) ring sort is a field.


By the basis theorem (and using the axiom of choice) every vector space admits a basis.

Revised on July 4, 2015 14:52:32 by Todd Trimble (