Mathematical Analysis An Introduction

This is a textbook suitable for a year-long course in analysis at the ad­ vanced undergraduate or possibly beginning-graduate level. It is intended for students with a strong background in calculus and linear algebra, and a strong motivation to learn mathematics for its own sake. At this stage of their education, such students are generally given a course in abstract algebra, and a course in analysis, which give the fundamentals of these two areas, as mathematicians today conceive them. Mathematics is now a subject splintered into many specialties and sub­ specialties, but most of it can be placed roughly into three categories: al­ gebra, geometry, and analysis. In fact, almost all mathematics done today is a mixture of algebra, geometry and analysis, and some of the most in­ teresting results are obtained by the application of analysis to algebra, say, or geometry to analysis, in a fresh and surprising way. What then do these categories signify? Algebra is the mathematics that arises from the ancient experiences of addition and multiplication of whole numbers; it deals with the finite and discrete. Geometry is the mathematics that grows out of spatial experience; it is concerned with shape and form, and with measur­ ing, where algebra deals with counting.

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.