Exercise 6, § 1

a) Let E be a complex Banach space and let $u\in \mathscr{L}(E)$. If there exists $n\in \mathbf{N}$ such that $n\geqslant 1$ and $u^n= 1_E$, then the spectrum of $u$ is contained in the group of $n$-th roots of unity and is formed of eigenvalues of $u$. b) Let G be a commutative topological group such that every element of G is of finite order. Every continuous irreducible representation of G in a Banach space is of dimension 1, and takes its values in the group $\mathbf{U}\subset \mathbf{C}^*$ of complex numbers of modulus 1.

c) Let G be a commutative topological group such that the set of elements of finite order of G is dense in G. Every continuous irreducible representation of G in a Banach space is of dimension 1, and takes its values in the group $\mathbf{U}\subset \mathbf{C}^*$ of complex numbers of modulus 1.