P-adic analysis ( p -adic analysis) is a branch of mathematics that deals with the mathematical analysis of functions of p-adic numbers.

The theory of complex-valued numerical functions on the p -adic numbers is just part of the theory of locally compact groups. The usual meaning taken for p -adic analysis is the theory of p -adic-valued functions on spaces of interest.

P-adic analysis is mainly applied 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.

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