Let be a field and be a non-empty set. Let be the set of all functions together with addition, defined by

, and all ,

and scalar multiplication defined by

, and .

Prove that is a vector space over by verifying each axiom in the definition.

You will need to refer to the axioms of the field . It is important to make a distinction between a function and one of its values .

The main issue I'm having with this one is wording, and the fact that I just got back into this after the summer, so I'm virtually brain-dead until I can get settled in again. The fact that I only just started learning anything about algebra-related fields doesn't help. Any help on this would be appreciated.