# Thread: Prove even functions set is a vector space

1. ## Prove even functions set is a vector space

Prove that set of even functions under addition and scalar multiplication is a vector space.

Proof. Let F denote the set of functions f with f(t) = f(-t) for all t.

Now let f and g be in F, then (f+g) (t) = f(t) + g(t), (fog)(t) = f(g(t)).

Is that enough to prove that it is a vector space?

Prove that set of even functions under addition and scalar multiplication is a vector space.

Proof. Let F denote the set of functions f with f(t) = f(-t) for all t.

Now let f and g be in F, then (f+g) (t) = f(t) + g(t), (fog)(t) = f(g(t)).

Is that enough to prove that it is a vector space?
If $f,g$ are even then $f+g$ is even and $kf$ is even where $k\in \mathbb{R}$. So it is a vector space.

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### set of even functions vector space

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