P-adic analysis

P-adic analysis

In mathematics, "p"-adic analysis is a branch of number theory 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.

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.

See also

* Mahler's theorem, which treats a "p"-adic analog of Taylor series.
* Hensel's lemma
* Locally compact space
* Real analysis


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