I would like to understand whether or not the following subspace of $L^2{[0,\infty)}$ is also a Hilbert space.
Let $H[0,\infty]$ be a subspace of $L^2{[0,\infty)}$, with inner product $\langle f, g\rangle=\int_{0}^{\infty}f(x)g(x)dx$, with the added conditions:
- for every $f\in H, f(x)=f(1/x)$
- $f(0)=0$. I am not sure if this condition is strictly needed as condition 1 combined with the fact that this is a subspace of $L^2$ should ensure that $f(0)=0$.
I am unsure how to proceed with the proof.
My initial instinct is to try to show that $H$ is a complete subspace of $L^2{[0,\infty)}$ and hence a Hilbert space but don't know how to properly deal with Cauchey Sequences. Is this the correct approach here?
Your space $H$ is closed and thus complete, since $L^2$ is itself complete, i.e. conditions 1 and 2 pass to limits.
The trick here is the following measure-theoretic lemma: if $(f_n)\subset L^p$ is a sequence of functions s.t. $\|f_n-f\|_{L^p}\to0$, then there exists a subsequence $(f_{n_k})$ of $(f_n)$ that converges to $f$ almost everywhere.
Thus if $(f_n)\subset H$ and $f_n\to f\in L^2$ for some $f\in L^2$, then pass to a subsequence $f_{n_k}$ that converges a.e. to $f$ and thus $$f(x)=\lim_kf_{n_k}(x)=\lim_kf_{n_k}(\frac{1}{x})=f(\frac{1}{x}) $$ almost everywhere.
Condition 2 in general doesn't make any sense since elements of $L^2$ are formally not functions but equivalence classes of a.e. equal functions, but if you were working say with $C_0(\mathbb{R})\cap L^2(\mathbb{R})$ then the same lemma would work to prove condition 2.
You can find a proof of this lemma in Folland's book for example.