Is the set of all periodic functions with similar period, a vector space

Question

Let $S$ be the set of all the periodic function with period $T$, is it a vector space?

I know a vector space is a set that is closed under finite vector addition and scalar multiplication, I want to prove that $S$ is a vector space (I believe there's no counter-example)

Answer 1

If $f,g \in S$ and if $\alpha$ is a scalar then we have

$(f+g)(x+T)=f(x+T)+g(x+T)=f(x)+g(x)=(f+g)(x)$

and

$(\alpha f)(x+T)=\alpha f(x+T)= \alpha f(x)= (\alpha f)(x).$

Conclusion ?