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基本説明
Based on the courses given by the author at Heidelberg. Covers the convergence of sequences, topological concepts including continuity, compactness and connectedness, differentiation in one variable, and more.
Full Description
Analysis I is part one of the undergraduate 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 functional analysis, measure theory and differential geometry. The book covers ""The convergence of sequences, topological concepts including continuity, compactness and connectedness, Resp. differentiation in one variable, theorems of Arzela-Ascoli and Stone-Weierstra[beta] and analytic functions in several variables, as well as Riemann integral. This book, which demands minimum prerequisites, is intended for first year graduate students or undergraduates who want to pursue the Math or Physics fields.
Contents
1. Foundations (Elements of Logic, Elements of set theory, Cartesian Product, Functions and Relations, Natural and Real Numbers); 2. Convergence (Convergence in R, Infinite series in R, Convergence in Rn, Metric spaces, Series in Banach spaces, Uniform convergence, Complex numbers); 3. Continuity (Topological concepts, Continuous maps, Compactness, The Tietze-Urysohn extension theorem, Connectedness, Product spaces, Continuous linear maps, Semicontinuous functions); 4. 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 - Ax1 The elementary functions, Polynomials, Taylor's formula); 5. Spaces of continuous functions (Dini's theorem, Arzela-Ascoli Theorem, The Stone-Weierstra[beta] Theorem, Analytic functions); 6. 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)
