A for a field, a vector space over is module over the ring . Sometimes a vector space over is called a -linear space. (Compare ‘-linear map’.)
The category of vector spaces is typically denoted Vect, or if we wish to make the field explicit. So
Vect_k \coloneqq k Mod
This category has vector spaces over as objects, and -linear maps between these as morphisms.
By the basis theorem (and using the axiom of choice) every vector space admits a basis.
Revised on January 9, 2013 15:06:27
by Urs Schreiber