# Subspaces of functions

1. Sep 13, 2005

### gaborfk

Yet another problem I need to get some starting help on:

Show that the set of continuous functions f=f(x) on [a,b] such that $$\int \limits_a^b f(x) dx=0$$ is a subspace of C[a,b]
Thank you

2. Sep 13, 2005

### NateTG

I would start by checking the definition of subspace.

3. Sep 13, 2005

### gaborfk

Definition of subspace means that the functions are closed under addition and scalar multiplication

4. Sep 13, 2005

### NateTG

So can you show that that's true for the potential subspace in your example?

5. Sep 13, 2005

### gaborfk

You mean that if $$\int \limits_a^b f(x) dx=0$$ and $$\int \limits_a^b g(x) dx=0$$, can I prove that $$\int \limits_a^b f(x)+g(x) dx=0$$? Also, if $$\int \limits_a^b f(x) dx=0$$ then $$k\int \limits_a^b f(x) dx=0$$?

6. Sep 13, 2005

### NateTG

Yeah, that's pretty much it. (Technically you also have to show that it's a subset, but in this case that's trivial.)

7. Sep 13, 2005

### gaborfk

Thank you!

The "hard ones" are so easy sometimes.....