Homeomorphism in periodic Sobolev space

Question

How can I show that between $H^{s}(T)$, and $l_{s}^{2}(\mathbb{Z})$ (the set of sequences $\mathbb{Z}\rightarrow \mathbb{C}$ $\mathbb{Z}\rightarrow \mathbb{C}$ such that $\sum_{k\in \mathbb{Z}}(1+\left | k \right |^{2})^{s}\left | \alpha (k) \right |^{2}< \infty $) there is a linear isometric homeomorphism?

Answer 1

Define $U : H^s(\mathbb{T})\rightarrow\ell^2_s(\mathbb{Z})$ by $$ Uf = \left\{ \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ikx}dx\cdot(1+ik)^{s} \right\}_{k=-\infty}^{\infty}. $$