Consider the Hilbert Space $\displaystyle H = L^2 (-1,1)$ of Lebesgue integrable real valued function on $\displaystyle (-1,1)$ with inner product $\displaystyle (f,g) = \int_{(-1,1)} fg dm$.

Prove that the set of all even function in $\displaystyle H$ is a closed subspace of $\displaystyle H$.