First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately:


Let $\varphi:\mathbb{Q}\_p\rightarrow\mathbb{C}$. 


Canonically, I would define a derivative of such a function in the following way:
$\varphi'(x):=\lim\limits\_{h\rightarrow 0} \frac{\varphi(x+h)-\varphi(x)}{|h|\_p}, $


provided of course that the limit exists. Is this the usual definition of the derivative?


In real functional analysis, there is the notion of Schwartz functions, which are rapidly decreasing functions supported on $\mathbb{R}^n$, i.e. functions, where for multiindices $\alpha,\beta$


$\sup\_{x\in\mathbb{R}^n}|x^\alpha D^\beta (f)(x)|<\infty.$


To my knowledge, the term Bruhat-Schwartz function generalizes this to the case where the domain is a locally compact abelian group, e.g. the field of $p$-adic numbers. Thus there should be an analogue notion where one considers $p$-adic derivatives.


Now I found in some sources, that a Bruhat-Schwartz function on $\mathbb{Q}\_p^n$ is a locally constant function with compact support. Unfortunately, I don't see at all how you get from the one definition to the other. 



Bruhat-Schwartz functions and derivatives in p-adic numbers
There is no good (nontrivial) notion for a derivative on functions on a totally disconnected group $G$ with values in $\mathbb{R}$ or $\mathbb{C}$. Assume (wlog) that $G$ is compact, then there exists a countable, ordered set of open, normal subgroups $(N\_i)$ with the projective limit of $G/N\_i$ being canonically isomorphic to $G$. So $G$ is a projective limit of finite groups. More generally this works for locally compact almost connected groups and Lie groups instead of finite groups. This was the motivation for Bruhat to introduce this notion for p-adic groups.


Another natural reason for why we say this is the analogue of a Schwartz space is that the Fourier trandform on $\mathbb{Q}\_p$ transforms compactly supported and being locally constant into each other. 


I suggest to have a look at Paul Sally survey on complex functions on $p$-adic groups.
<http://link.springer.com/article/10.1023%2FA%3A1007583108067#page-1>


