Corollary 1

Let K be a locally compact non-discrete field not necessarily commutative and $\chi$ a nontrivial unitary character of the additive group of K. Let E be a finite-dimensional topological vector space over K. The map $\theta$ of $E\times E'$ into $\mathbf{U}$ defined by $\theta (x, \lambda ) =\chi (\langle \lambda , x\rangle )$ for $(\lambda , x)\in E'\times E$ puts the topological groups E and $E'$ in duality.

Let $n$ be the dimension of E and $(e_1, . . . , e_n)$ a basis of E. It makes it possible to identify E and $K^n$ (EVT, I, p. 14, Theorem 2). The result then follows from Proposition 19 and Proposition 2 of II, p. 206.

We denote by $\mathbf{T}$ the group $\mathbf{R}/\mathbf{Z}$.