# Subspaces of functions

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

## Answers and Replies

Science Advisor
Homework Helper
I would start by checking the definition of subspace.

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

Science Advisor
Homework Helper
gaborfk said:
Definition of subspace means that the functions are closed under addition and scalar multiplication

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

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 $$k\int \limits_a^b f(x) dx=0$$?

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

gaborfk
Thank you!

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