Editing P-adic analysis (section)

{{Short description|Branch of number theory}}
{{DISPLAYTITLE:''p''-adic analysis}}
[[Image:3-adic integers with dual colorings.svg|thumb|The 3-adic integers, with selected corresponding characters on their [[Pontryagin dual]] group]]
In [[mathematics]], '''''p''-adic analysis''' is a branch of [[number theory]] that studies functions of [[p-adic number|''p''-adic numbers]]. Along with the more classical fields of [[real analysis|real]] and [[complex analysis]], which deal, respectively, with functions on the [[real numbers|real]] and [[complex numbers|complex]] numbers, it belongs to the discipline of [[mathematical analysis]].

The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of [[locally compact group]]s ([[abstract harmonic analysis]]). The usual meaning taken for ''p''-adic analysis is the theory of ''p''-adic-valued functions on spaces of interest.

Applications of ''p''-adic analysis have mainly been in [[number theory]], where it has a significant role in [[diophantine geometry]] and [[diophantine approximation]]. Some applications have required the development of ''p''-adic [[functional analysis]] and [[spectral theory]]. In many ways ''p''-adic analysis is less subtle than [[classical analysis]], since the [[ultrametric inequality]] means, for example, that convergence of [[infinite series]] of ''p''-adic numbers is much simpler. [[Topological vector space]]s over ''p''-adic fields show distinctive features; for example aspects relating to [[convex set|convexity]] and the [[Hahn–Banach theorem]] are different.