Exercise 31, § 1

Let $r\geqslant 1$ be an integer. We denote by $\mathscr{S}(\mathbf{T}^r)$ the complex vector space of indefinitely differentiable functions on $\mathbf{T}^r$, and by $\mathscr{S}(\mathbf{Z}^r)$ the complex vector space of functions $f:\mathbf{Z}^r\rightarrow \mathbf{C}$ such that, for every real A $>0$, one has $f(n) = O(\|n\|^{-A})$ when $\|n\| \rightarrow +\infty$.

a) Endowed with the semi-norms

$p_A(f) =$ sup$_{n\in\mathbf{Z}^r}\|n\|^A|f(n)|$,

the space $\mathscr{S}(\mathbf{Z}^r)$ is a Fréchet space.

b) For every real $p\in [1,+\infty [$, the space $\mathscr{S}(\mathbf{Z}^r)$ (resp. the space $\mathscr{S}(\mathbf{T}^r)$) is dense in $L^p(\mathbf{Z}^r)$ (resp. in $L^p(\mathbf{T}^r)$).

c) The restriction of the Fourier transform of $\mathbf{T}^r$ to $\mathscr{S}(\mathbf{T}^r)$ is an isomorphism of $\mathscr{S}(\mathbf{T}^r)$ onto $\mathscr{S}(\mathbf{Z}^r)$, whose inverse is the restriction of the inverse Fourier transform.

d) The topological vector space $\mathscr{S}(\mathbf{Z}^r)$ is isomorphic to $\mathscr{S}(\mathbf{R}^n)$, in particular it is a Montel space. (Use exercise 3.)