# Vector space

1. Feb 25, 2009

### ak123456

1. The problem statement, all variables and given/known data
Let S be a set and let V be a vector space over the field F. LetV^S denote the set of all maps from S to V . We define an addition on V^S and a scalar multiplication of F on V^S as follows: (f+g)(s):=f(s)+g(s) and (af)(s):=a(f(s)) for any s belongs to S
show that V^S is a vector space over F

2. Relevant equations

3. The attempt at a solution
how to show it is a vector space , do i need to show all the axioms exist ? or ,is there any specific way to use ?

Last edited: Feb 25, 2009
2. Feb 25, 2009

### HallsofIvy

Staff Emeritus
Well, you should be able to assume such things as associativity of addition and scalar multiplication- they are true of any mapping. But you will need to show explicitely that the sum of two such mappings is again a mapping and that the scalar product of a number and a mapping is a mapping.

3. Feb 25, 2009

### ak123456

so i have to show all the axioms for Vector Space ??