Analysis /
Analysis I is part one of the international series in analysis. This book is based on the courses given by the author at Heidelberg. It comprises of materials for a one and a half semester, and can be used as a textbook. The contents range from elementary calculus to fairly advanced topics in functi...
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| Format: | Book |
| Language: | English |
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Cambridge, MA :
International Press,
[2003-]
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| Series: | Graduate series in analysis.
International series in analysis. |
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Table of Contents:
- Chapter 0: Foundations (Elements of Logic, Elements of set theory, Cartesian Product, Functions and Relations, Natural and Real Numbers)
- Convergence (Convergence in R, Infinite series in R, Convergence in Rn, Metric spaces, Series in Banach spaces, Uniform convergence, Complex numbers)
- Continuity (Topological concepts, Continuous maps, Compactness, The Tietze-Urysohn extension theorem, Connectedness, Product spaces, Continuous linear maps, Semicontinuous functions)
- Differentiation in one Variable (Differentiable functions, The mean value theorem and its consequences, De L'Hospital's Rule, Differentiation of sequences of functions, The differential equation x' = Ax, The elementary functions, Polynomials, Taylor's formula)
- Spaces of continuous functions (Dini's theorem, Arzela-Ascoli Theorem, The Stone-Weierstra Theorem, Analytic functions)
- Integration in one variable (The Riemann integral, Integration rules, Monotone and continuous functions are integrable, Fundamental theorem of calculus, Integral theorems and transformation rules, Integration of rational functions, Lebesgue's integrability criterion, Improper integrals, Parameter dependent integrals)