# Subspaces of functions

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

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

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

NateTG
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?

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

NateTG