# Invertible map on a particular sequence space

For $$a\in\mathbb R$$, let $$h_a$$ be the Hilbert space of sequences defined by $$h_a=\left\{(x_n):\sum_{n\in\mathbb Z}(1+n^2)^a|x_n|^2<\infty\right\}$$ and inner product $$\langle(x_n),(y_n)\rangle_a=\sum_{n\in\mathbb Z}(1+n^2)^ax_n\overline{y_n}$$.

Define the function $$f:h_{-a}\to(h_a)^*$$ by $$f\big((x_n)\big)((y_n))=\sum_{n\in\mathbb Z}x_ny_n,$$ where $$(x_n)\in h_{-a}$$ and $$(y_n)\in h_a$$. Prove that

1. $$f((a_n))$$ is well-defined as a function on $$(h_a)^*$$.
2. $$f$$ is an invertible, continuous linear map and has a bounded inverse.

Attempt: I have shown already that the series $$\sum x_ny_n$$ is convergent: if $$(x_n)\in h_{-a}$$ and $$(y_n)\in h_a$$ then $$\sum(1+n^2)^{-a}|x_n|^2<\infty,\qquad\sum(1+n^2)^a|y_n|^2<\infty,$$ so by Cauchy-Schwarz I have \begin{aligned} \left|\sum|x_ny_n|\right|^2&=\left|\sum((1+n^2)^{-a/2}|x_n|)((1+n^2)^{a/2}|y_n|)\right|^2\\ &\leq\left(\sum\left|((1+n^2)^{-a/2}|x_n|)^2\right|\right)\left(\sum\left|((1+n^2)^{a/2}|y_n|)^2\right|\right)\\ &<\infty, \end{aligned} which means $$\sum x_ny_n$$ is convergent.

To finish showing that $$f$$ is a function from $$h_{-a}$$ to $$(h_a)^*$$, I think that I need to show somehow that $$f((x_n))$$ is a continuous linear functional. That $$f$$ is linear is clear to me, but I don't know how to show it is continuous. I know that one way to show continuity is to show boundedness, but I also don't know how to proceed in that direction. As for Part (2), I am unsure how to begin as well; in particular, I'm having trouble visualizing what the inverse map from $$(h_a)^*$$ to $$h_{-a}$$ would look like.

Any help or hints on this problem would be greatly appreciated. Thank you in advance.

You have proved that $$|f((x_n)) ((y_n))| \leq \|(x_n)\|\|(y_n)\|$$ where the norms are taken the appropriate spaces. This implies that $$\|f((x_n))\|\leq \|(x_n))\|$$ and hence $$f$$ is abounded operator with $$\|f\|\leq 1$$.
Hints for then second part: You know that $$f$$ is injective. To show that $$f$$is surjective use the fact that ant element of $$h_a^{*}$$ is given by an inner product with some element of $$h_a$$. This follows by Riesz Theorem. Completeness of the spaces $$h_a$$ can be proved exactly the way you prove completeness of $$\ell^{p}$$ spaces. Thus each $$h_a$$ is a Hilbert space and Riesz Theorem is applicable.