How can I show that the subset of $L^2(-a, a)$ of the functions with mean value 0 is Hilbert?

Question

As in the title, let's define the set as \begin{equation} \mathcal{F} = \{f \in L^2(-a,a) | \int_{-a}^a{f(x)dx}=0 \} \end{equation} How can I show this is a Hilbert subspace? I noticed that every finite linear combination of normalized functions from this set has still mean value 0. Is it correct to deduce that any infinite combination has mean value 0?

Answer 1

You already know that $\mathcal F$ is a linear subspace. What remains is to show that it closed in $L^{2}$. If $f_n \in \mathcal F$ for all $n$ and $f_n \to f$ in $L^{2}$ then $\int_{-a}^{a} f(x)dx=0$ because $\int_{-a}^{a} f_n(x)dx=0$ and $|\int_{-a}^{a} f(x)dx-\int_{-a}^{a}f_n(x)dx| \leq \sqrt {2a} \sqrt {\int_{-a}^{a} |f_n-f|^{2}} \to 0$ (where I have used Cauchy Scawarz inequality).