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).

  1. ,
  2. ,
  3. ,
  4. For every , there exists a such that ,
  5. ,
  6. ,
  7. ,
  8. .