The theory of complex-valued numerical functions on the p-adic numbers is 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.
- Mahler's theorem, which treats a p-adic analog of Taylor series.
- Hensel's lemma
- Locally compact space
- P-adic quantum mechanics
- Real analysis
- Koblitz, Neal (1980). p-adic analysis: a short course on recent work. London Mathematical Society Lecture Note Series 46. Cambridge University Press. ISBN 0-521-28060-5. Zbl 0439.12011.
- Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts 3. Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006.
- A course in p-adic analysis, Alain Robert, Springer, 2000, ISBN 978-0-387-98669-2
- Ultrametric Calculus: An Introduction to P-Adic Analysis, W. H. Schikhof, Cambridge University Press, 2007, ISBN 978-0-521-03287-2
- P-adic Differential Equations, Kiran S. Kedlaya, Cambridge University Press, 2010, ISBN 978-0-521-76879-5
|This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.|