Mathematical Analysis

This book introduces analysis, from the basics of mathematical proposition and proof to differentiation, integration and more. Includes Stokes theorem and such applications as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions.

This is a textbook suitable for a year-long course in analysis at the advanced undergraduate or beginning graduate level. It is intended for students with a strong background in calculus and linear algebra. The topics covered in this course are the calculus of functions of one variable, an introduction to general topology, the general theory of integration, and the calculus, differential and integral, for functions of several variables.

Inhalt
1 Real Numbers.- 1.1 Sets, Relations, Functions.- 1.2 Numbers.- 1.3 Infinite Sets.- 1.4 Incommensurability.- 1.5 Ordered Fields.- 1.6 Functions on R.- 1.7 Intervals in R.- 1.8 Algebraic and Transcendental Numbers.- 1.9 Existence of R.- 1.10 Exercises.- 1.11 Notes.- 2 Sequences and Series.- 2.1 Sequences.- 2.2 Continued Fractions.- 2.3 Infinite Series.- 2.4 Rearrangements of Series.- 2.5 Unordered Series.- 2.6 Exercises.- 2.7 Notes.- 3 Continuous Functions on Intervals.- 3.1 Limits and Continuity.- 3.2 Two Fundamental Theorems.- 3.3 Uniform Continuity.- 3.4 Sequences of Functions.- 3.5 The Exponential function.- 3.6 Trigonometric Functions.- 3.7 Exercises.- 3.8 Notes.- 4 Differentiation.- 4.1 Derivatives.- 4.2 Derivatives of Some Elementary Functions.- 4.3 Convex Functions.- 4.4 The Differential Calculus.- 4.5 L'Hospital's Rule.- 4.6 Higher Order Derivatives.- 4.7 Analytic Functions.- 4.8 Exercises.- 4.9 Notes.- 5 The Riemann Integral.- 5.1 Riemann Sums.- 5.2 Existence Results.- 5.3 Properties of the Integral.- 5.4 Fundamental Theorems of Calculus.- 5.5 Integrating Sequences and Series.- 5.6 Improper Integrals.- 5.7 Exercises.- 5.8 Notes.- 6 Topology.- 6.1 Topological Spaces.- 6.2 Continuous Mappings.- 6.3 Metric Spaces.- 6.4 Constructing Topological Spaces.- 6.5 Sequences.- 6.6 Compactness.- 6.7 Connectedness.- 6.8 Exercises.- 6.9 Notes.- 7 Function Spaces.- 7.1 The Weierstrass Polynomial Approximation Theorem . . ..- 7.2 Lengths of Paths.- 7.3 Fourier Series.- 7.4 Weyl's Theorem.- 7.5 Exercises.- 7.6 Notes.- 8 Differentiable Maps.- 8.1 Linear Algebra.- 8.2 Differentials.- 8.3 The Mean Value Theorem.- 8.4 Partial Derivatives.- 8.5 Inverse and Implicit Functions.- 8.6 Exercises.- 8.7 Notes.- 9 Measures.- 9.1 Additive Set Functions.- 9.2 Countable Additivity.- 9.3Outer Measures.- 9.4 Constructing Measures.- 9.5 Metric Outer Measures.- 9.6 Measurable Sets.- 9.7 Exercises.- 9.8 Notes.- 10 Integration.- 10.1 Measurable Functions.- 10.2 Integration.- 10.3 Lebesgue and Riemann Integrals.- 10.4 Inequalities for Integrals.- 10.5 Uniqueness Theorems.- 10.6 Linear Transformations.- 10.7 Smooth Transformations.- 10.8 Multiple and Repeated Integrals.- 10.9 Exercises.- 10.10 Notes.- 11 Manifolds.- 11.1 Definitions.- 11.2 Constructing Manifolds.- 11.3 Tangent Spaces.- 11.4 Orientation.- 11.5 Exercises.- 11.6 Notes.- 12 Multilinear Algebra.- 12.1 Vectors and Tensors.- 12.2 Alternating Tensors.- 12.3 The Exterior Product.- 12.4 Change of Coordinates.- 12.5 Exercises.- 12.6 Notes.- 13 Differential Forms.- 13.1 Tensor Fields.- 13.2 The Calculus of Forms.- 13.3 Forms and Vector Fields.- 13.4 Induced Mappings.- 13.5 Closed and Exact Forms.- 13.6 Tensor Fields on Manifolds.- 13.7 Integration of Forms in Rn.- 13.8 Exercises.- 13.9 Notes.- 14 Integration on Manifolds.- 14.1 Partitions of Unity.- 14.2 Integrating k-Forms.- 14.3 The Brouwer Fixed Point Theorem.- 14.4 Integrating Functions on a Manifold.- 14.5 Vector Analysis.- 14.6 Harmonic Functions.- 14.7 Exercises.- 14.8 Notes.- References.