Given $F$ a Local field which is also a locally compact abelian group, we donote the Schwartz-Bruhat functions on it by $\mathcal{S}(F)$. Without using direct limit language and after simple computation, we get any $f \in \mathcal{S}(F)$ can be represented by finite $\mathbb{C}-$ combination of $1_{U}$, $U$ is a compact open set in $F$.
Now, how can I define the norm, or convergence in the space $\mathcal{S}(F)$? Since $\mathcal{S}(F)$ is dense in $L^2(F)$, I suppose that the norm will be $\|\cdot\|_2$. However, I met difficulty in proving Fourier Transform $\mathcal{F}$ continuous using this norm.(Suppose we chose standard character and self-dual measure w.r.t the standard character)
Thanks for any help!
For $F$ a finite extension of $\Bbb{Q}_p$ or $\Bbb{F}_p((t))$
$S(F)$ can't be a complete metric space (assume the opposite, take $c_k>0$ decreasing fast enough so that $\sum_{k\ge 1} c_k 1_{\pi_F^k O_F} $ converges in the metric)
The convergence should be "$f_n\to f$ iff for some $a,b\in F^*$, the $f_n,f$ are supported on a common compact $aO_F$, are $b O_F$ invariant, and converge pointwise"
So let $S(F,a,b)$ be the space of functions $F\to \Bbb{C}$ supported on $aO_F$ and $bO_F$ invariant, which is quite the same as the functions $aO_F/(bO_F\cap aO_F) \to \Bbb{C}$. The $\sup$ norm makes it complete and $S(F)=\bigcup_{a,b\in F^*} S(f,a,b)$. Then $U$ is open in $S(F)$ iff $U\cap S(F,a,b)$ is open in $S(F,a,b)$ for all $a,b\in F^*$.