Exercise 22, § 1

Let $p$ be a prime number. Let K be a finite extension of the field $\mathbf{Q}_p$ of $p$-adic numbers; it is a locally compact non-discrete field. Let $t: K\rightarrow \mathbf{Q}_p$ be the trace mapping. Let $\lambda :\mathbf{Q}_p\rightarrow \mathbf{Q}$ be the mapping defined in Prop. 20 of II, p. 237.

a) The additive group of K is in duality with itself by means of the mapping $(x, y)\mapsto$ exp(2$i\pi \lambda (t(xy))$). We identify K and its dual by this mapping.

For every integer $n\geqslant 1$, we denote by $\mathscr{S}(K^n)$ the complex vector space of functions from $K^n$ into $\mathbf{C}$ which are locally constant and of compact support. b) The restriction of the Fourier transform to $\mathscr{S}(K^n)$ is an automorphism whose inverse is the restriction of the Fourier cotransformation.

c) For every real $p\in [1,+\infty [$, the space $\mathscr{S}(K^n)$ is dense in $L^p(K^n)$.