Are Continuous Functions with Zero Integral a Subspace of C[a,b]?

[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

 

[NateTG]

I would start by checking the definition of subspace.

 

[gaborfk]

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

 

[NateTG]

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

 

[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.)

 

[gaborfk]

Thank you!

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