P-adic analysis

In mathematics, p-adic analysis is a branch of number theory that studies functions of p-adic numbers. Along with the more classical fields of real and complex analysis, which deal, respectively, with functions on the real and 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 groups (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 spaces over p-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem are different.