Schikhof W.H. Ultrametric Calculus: An Introduction to p-adic Analysis

Cambridge University Press, 1984. — 318 p. — (Cambridge Studies in Advanced Mathematics, 4). — ISBN: 0-521-24234-7, 978-0-521-24234-9, 0-521-03287-3, 978-0-521-03287-2.This is an introduction to p-adic analysis which is elementary yet complete and which displays the variety of applications of the subject. Dr. Schikhof is able to point out and explain how p-adic and 'real' analysis differ. This approach guarantees the reader quickly becomes acquainted with this equally 'real' analysis and appreciates its relevance. The reader's understanding is enhanced and deepened by the large number of exercises included throughout; these both test the reader's grasp and extend the text in interesting directions. As a consequence, this book will become a standard reference for professionals (especially in p-adic analysis, number theory and algebraic geometry) and will be welcomed as a textbook for advanced students of mathematics familiar with algebra and analysis.Valuations
The strong triangle inequality
The p-adic integers
The p-adic numbers
Topological properties of Q_p
Q_p as a completion of Q
Q_p compared to R
Archimedean and non-archimedean valuations
Equivalence of valuations
All valuations on Q
The residue class field and the value group
Series expansions of elements of К
Normed spaces
Extensions of valuations·
Uniqueness of the extended valuation
The valuation on the algebraic closure
Completion of the algebraic closure. C_p
Ultrametrics
Ultrametric spaces
Compactness and separability
Spherical completeness
Best approximation
Calculus
Elementary calculus
The classical concepts of calculus
Sequences and series
Order-like structure in K
(Locally) analytic functions
Continuity and differentiability
Continuously differentiable functions
Twice continuously differentiable functions
Cⁿ-functions
Antiderivation and integration
Interpolation
The idea of interpolation
p-adic exponents
Roots of unity in C_p. The Teichmueller character
Σ^x_n=0 a_n for a p-adic integer χ
The p-adic gamma function
A p-adic Euler constant
Values of Γ_p in 1/2, 0, -1, -2...
The p-adic Gauss-Legendre multiplication formula
Some other formulas involving Γ_p
Analytic functions
Convergence of power series
Substitution of power series
The maximum principle
Failure of the maximum principle for locally compact K
exp and log
Extensions of exp and log
Trigonometric functions
(1+x)^a
The Artin-Hasse exponential
arcsin and arccos
Functions on Z_p
Mahler's base and p-adic integration
Orthogonal bases in Banach spaces
The Mahler base of С(Z_p→K)
The Mahler coefficients. Examples
Mahler's base for C¹(Z_p→K)
Mahler coefficients of Cⁿ-functions
The Volkenborn integral
The Bernoulli numbers
Integration over subsets
The p-adic gamma and zeta functions
Local analyticity of Γ_p
A formula for log_p2
Diamond's log gamma function
The p-adic zeta functions
van der Put's base and antiderivation
van der Put's base of C(Z_p→K).
Characterizations by means of coefficients
Antiderivation
The differential equation y' = F(x,y)
C1-solutions of a meromorphic differential equation
p-adic Liouville numbers
van der Put's base of C¹(Z_p→K)
More general theory of functions
Continuity and differentiability
Convergent sequences of differentiable functions
A function of the first class has an antiderivative
Points at which a differentiable function is С¹
Local behaviour of differentiable functions
Lusin-type theorems
Differentiable homeomorphisms
Isometries
Extension theorems
Cⁿ-theory
Local invertibility of Cn-functions
Differentiation Cⁿ→C^n-1
Antiderivation С→C¹
Antiderivation C^n-1→Cⁿ. A candidate
Surjectivity of differentiation Cⁿ→C^n-1
Surjectivity of differentiation C^∞→C^∞
C³-functions
Functions of two variables
Monotone functions
Sides of 0 in K
Monotone functions of type σ
Monotonicity without type
Aspects of functional analysis
Two theorems on metric spaces
Functions of the first class of Baire
Orthonormal bases of С(X→K)
The ultrametric Stone-Weierstrass theorem
Integration on compact spaces
Measures and distributions on Z_p
A substitution formula for real valued integrals
The ultrametric Hahn-Banach theorem
A field with prescribed residue class field and value group
Isometrical embedding of an ultrametric space into К
Glossary of terms
Sets
Subsets of R
Metric and topology
Algebra