# How to show set of functions is a vector space?

## Homework Statement

Let S be a set and V a vector space over the field k. Show that the set of functions f: S --> k, under function addition and multiplication by a constant is a vector space.

## Homework Equations

I think I need to show that (f+g)(x) = f(x) + g(x) and that (rf)(x) = rf(x)

## The Attempt at a Solution

Are the above conditions true because both functions f and g for this particular case give you k and (k1 + k2) = k1 + k2 and rk = rk. I just don't understand how to show it. Or do I just literally write (f+g)(x) = f(x) + g(x) and that (rf)(x) = rf(x).

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
Yes, that's exactly right. The set of all functions, with the standard definitions of "sum of two functions" and "product of a function and a number", is a vector space because for any two functions f and g, f+ g is also a function and for any function f and real number r, rf is a function.

Strictly speaking you should also show or at least state that addition of functions is associative, commutative, etc. but those might be assumed.