# Thread: Need a proof for a subspace

1. ## Need a proof for a subspace

Hi all,

The question goes: Let V be the set of all continous functions on the closed interval [a,b]. Prove that the set W of continuous functions f(x) on [a,b] such that integral(f(x)dx) = 0 from a to b is a subspace of V.

Does this involve utilizing the fundamental theorem of calculus in any way? Thanks!

2. Originally Posted by lute
Let V be the set of all continous functions on the closed interval [a,b]. Prove that the set W of continuous functions f(x) on [a,b] such that integral(f(x)dx) = 0 from a to b is a subspace of V.
You do know how one proves a set is a subspace?

If $\alpha$ is a scalar and $\left\{ {f,g} \right\} \subseteq W$ then what can you say about $\int_a^b {\alpha f + g}$ ?

3. Originally Posted by lute
Hi all,

The question goes: Let V be the set of all continous functions on the closed interval [a,b]. Prove that the set W of continuous functions f(x) on [a,b] such that integral(f(x)dx) = 0 from a to b is a subspace of V.

Does this involve utilizing the fundamental theorem of calculus in any way? Thanks!
No, it involves the definition of "subspace"! If f and g are two continuous functions such that $\int_a^b f(x)dx= 0$ and $\int_a^b g(x)dx$ is f+ g a continuous function such that $\int_a^b(f(x)+ g(x))dx= 0$? If n is a real number is nf(x) a continuouis function such that $\int_a^b nf(x)dx= 0$?