Corollary 4

The group $\mathbf{R}^*$ is in duality with the group $\{-1,1\} \times \mathbf{R}$ by the mapping $(x,(\sigma , t))\mapsto \sigma (x/|x|)|x|^{it}$. The group $\mathbf{R}^*_+$ is in duality with $\mathbf{R}$ by the mapping $(x, t)\mapsto x^{it}$.

In fact, the mapping $x\mapsto (x/|x|$, log($|x|$)) is a topological group isomorphism of $\mathbf{R}^*$ onto $\{-1,1\} \times \mathbf{R}$. The assertion then follows from lemma 7 of II, p. 225, from corollary 2, and from the fact that the unitary characters of $\{-1,1\}$ are 1 and $x\mapsto x$.

Let $p$ be a prime number. The field $\mathbf{Q}_p$ of $p$-adic numbers is the completion of $\mathbf{Q}$ for the $p$-adic valuation (INT, VII, § 1, No.$^o6$, example, and AC, VI, § 3, No.$^o4$, example 4). For every $x\in \mathbf{Q}_p$, there exist a unique integer $\nu \geqslant 0$ and a unique integer $q$ satisfying $0\leqslant q < p^{\nu}$ such that $qp^{-\nu}-x\in \mathbf{Z}_p$ (A, VII, p. 10, Theorem 2, applied to the principal ideal domain $\mathbf{Z}_p$ and to the set $R_p$ of integers $j$ such that $0\leqslant j < p$). We put $\lambda (x) =qp^{-\nu}$.