The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4.
The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define $\mathcal{H}_\eta$ to be the space of all measurable functions $f$ on $[a,b]$ such that
$$\int_a^b |f(t)|^2\eta(t)dt<\infty.$$
Define the inner product on $\mathcal{H}_\eta$ by $$\langle f, g\rangle_\eta=\int_a^bf(t)\overline{g(t)}\eta(t)dt.$$
Show that $\mathcal{H}_\eta$ is a Hilbert space, and that the mapping $U:f\mapsto \eta^{1/2}f$ gives a unitary correspondence between $\mathcal{H}_\eta$ and the usual space $L^2[a,b]$.
Discussion It is not hard to show that $\langle f, g\rangle_\eta$ gives an inner product on $\mathcal{H}_\eta$, and once it is established that $\mathcal{H}_\eta$ is a Hilbert space, showing that $U$ is a Unitary mapping is not too difficult either- I have been able to complete these steps in full. The difficulty I am having is completing the proof that $\mathcal{H}_n$ is actually a Hilbert space.
To do this I must show that $\mathcal{H}_\eta$ is a Banach space over $\mathbb{C}$ equipped with the norm $\|f\|=\sqrt{\langle f, f\rangle }_\eta$, In particular, it remains to show $(\mathcal{H}_\eta, \|\cdot \|)$ is complete and separable.
I have an idea and perhaps you guys could provide some feedback on it. Let $f\in \mathcal{H}_\eta$. Since $[a,b]$ is compact, and $\eta(t)$ is continuous on $[a,b]$, $\eta(t)$ achieves its minimum on $[a,b]$ and since $\eta$ is strictly positive, $$ c:=\min_{\xi\in [a,b]}\eta(\xi)>0.$$ The monotonicity of the integral then gives:
$$\infty>\|f\|_{\mathcal{H}_\eta}=\int_a^b |f(t)|^2\eta(t)dt\geq \min_{\xi\in [a,b]}\eta(\xi) \int_a^b |f(t)|^2dt =c\|f\|_{L^2[a,b]} \enspace (*)$$ This gives $f\in L^2[a,b]$. Let $\{f_n\}$ be a Cauchy sequence in $\mathcal{H}_\eta$, i.e. $$\lim_{n,m\to\infty} \|f_n-f_m\|_{\mathcal{H}}=0$$
By $(*)$ we also have $\|f_n-f_m\|_{L^2[a,b]}\to 0$ as $n\to \infty,m\to \infty$. Thus $\{f_n\}$ is a Cauchy sequence in $L^2[a,b]$ and since $L^2$ is complete there exists $f\in L^2[a,b]$ such that $\lim_{n\to\infty}\|f_n-f\|_{L^2[a,b]}=0$.
Observe now that $\eta(t)$ (being continuous on a compact interval) achieves its maximum $d:=\max_{\xi\in [a,b]}\eta(\xi)>0$ and $$\int_a^b |f(t)|^2\eta(t)dt\leq d\|f\|_{L^2[a,b]}<\infty\enspace (**)$$ so $f$ is also in $\mathcal{H}_n$ and we have by $(**)$ that $$\frac{1}{d}\|f_n-f\|_{\mathcal{H}_\eta}<\|f_n-f\|_{L^2[a,b]}\to 0$$ so $\lim_{n\to infty}\|f_n-f\|_{\mathcal{H}_\eta}=0$ and thus $\mathcal{H}_\eta$ is complete.
Is this correct? As for separability, I am not as confident. I am guessing I can use a similar density argument since $L^2[a,b]$ is separable.
Thanks!