I was wondering whether we can say something about different inner product spaces of the form:
$L_\omega^2[a,b]= \Big \{f:[a,b]\rightarrow \mathbb{R}: \langle f, f\rangle_\omega<\infty \Big\}$ where $\omega:[a,b]\rightarrow \mathbb{R}$ is an almost sure positive function (with respect to the Lebesgue measure) such that $\langle 1,1 \rangle_{\omega}$ is finite.
Are there conditions under which we can say for two such "weight" functions, $\omega_1, \omega_2$, that $L_{\omega_1}^2[a,b]\subseteq L_{\omega_2}^2[a,b]$?
This stems from a particular case which I encountered where $[a,b]=[-1,1]$, $\omega_1(x)\equiv 1$ and $\omega_2(x)=\frac{1}{\sqrt{1-x^2}}$. I was wondering whether every $f\in L^2[-1,1]$ can be written as a generalized Fourier series with respect to an orthonormal basis in $L^2_{\frac{1}{\sqrt{1-x^2}}}[-1,1]$?
$L_{\omega_1}^2[a,b]\subseteq L_{\omega_2}^2[a,b]$ iff $w_2 \leq C w_1$ almost everywhere for some finite constant $C$. Proof. Using Closed Graph Theorem we can see easily that the inclusion map is continuous. [You will have to use the fact that convergence in the these norms imply almost sure convergence for a subsequence]. Hence $\|f\|_{L_{\omega_2}^2[a,b]} \leq C \|f\|_{L_{\omega_2}^2[a,b]} $ for some $C$ and for all $f$ which implies that $w_2 \leq C w_1$ almost everywhere