Exercise 7, § 1

For every integer $n\geqslant 1$, let $s_n$ denote the function $\mathbf{Z}\rightarrow \mathbf{R}$ defined by

$(1-k)^{n(1-k)}$ if $k\leqslant -1$

$s_n(k) =$ 1 if $k= 0$

$(k+ 1)^{-(k+1)/n}$ if $k\geqslant 1$.

Let S denote the set of the functions $s_n$.

a) For every integer $n\geqslant 1$ and every $(k_1, k_2)\in \mathbf{Z}^2$, one has $s_{4n}(k_1)s_{4n}(k_2)\geqslant$ $s_n(k_1+k_2)$.

b) For $f\in \mathbf{C}$(T), let $(a_k(f))_{k\in\mathbf{Z}}$ denote the coefficients of the Laurent expansion of $f$ at 0. For every $s\in S$ and every $f\in \mathbf{C}$(T), the series

converges, and the sets

defined for $s\in S$ and $\varepsilon >0$, form a fundamental system of open neighbourhoods of 0 for a locally convex topology on $\mathbf{C}(T)$. The multiplication $\mathbf{C}(T)\times \mathbf{C}(T)\rightarrow \mathbf{C}(T)$ is continuous for this topology.

c) The topology defined in the preceding question is metrizable but is not complete.

d) For $f\in \mathbf{C}(T)^*$, let $\varrho (f)$ denote the mapping $g\mapsto f g$ of $\mathbf{C}(T)$ into $\mathbf{C}(T)$. The mapping $\varrho$ is a continuous representation of $\mathbf{C}(T)^*$ in the locally convex space $\mathbf{C}(T)$.

e) The representation $\varrho$ is irreducible.