# Vector space/Direct/Definition

Vector space

Let denote a field, and a set with a distinguished element , and with two mappings

and

Then is called a
-* vector space*
(or a vector space over ),
if the following axioms hold
(where
and
are arbitrary).

- ,
- ,
- ,
- For every , there exists a such that ,
- ,
- ,
- ,
- .