Canonical isometric isomorphism of $l_{\alpha}^{2}$

Question

Let $\alpha \in \mathbb{R}$ and $l_{\alpha}^{2}$ the vector space of bi-infinite sequences $(x_{n})_{n\in \mathbb{Z}}$ such that $||x||_{\alpha}:=\sum_{n\in\mathbb{Z}} (1+n^{2})^{\alpha}|x_{n}|^2<\infty $. I've shown, that $(l_{\alpha}^{2},||.||_{\alpha})$ is Hilbert. But how can i describe a canonical isometric isomoprhism $(l_{\alpha}^{2})^{*}\cong l_{-\alpha}^{2}$?

Answer 1

If $y\in \ell_{-\alpha}^2$, define $T_y(x):=\sum_{j\in\mathbf Z}x_jy_j$ for $x\in \ell_\alpha^2$. This is well defined and $T_y$ is a linear continuous functional of norm $\lVert y\rVert_{\ell^2_{-\alpha}}$.

Actually, consider the map $M\colon (x_j)_{j\in\mathbb Z}\in \ell_\alpha^2\mapsto ((1+j^2)^{\alpha/2}x_j)_{j\in\mathbf Z}\in\ell^2$. This is an isometry. Then consider the adjoint map.