RAA Contents

Real Analysis and Applications

Table of Contents



Part A		Analysis
	1	Review	3
	1.1	Calculus	3
	1.2	Linear Algebra	5
	1.3	Appendix: Equivalence Relations	7

	2	The Real Numbers	9
	2.1	An Overview of the Real Numbers	9
	2.2	The Real Numbers and Their Arithmetic	10
	2.3	The Least Upper Bound Principle	13
	2.4	Limits	15
	2.5	Basic Properties of Limits	19
	2.6	Monotone Sequences	20
	2.7	Subsequences	23
	2.8	Cauchy Sequences	27
	2.9	Countable Sets	31

	3	Series	35
	3.1	Convergent Series	35
	3.2	Convergence Tests for Series	39
	3.3	Absolute and Conditional Convergence	44

	4	The Topology of R^n	48
	4.1	n-dimensional Space	48
	4.2	Convergence and Completeness in R^n	52
	4.3	Closed and Open Subsets of R^n	56
	4.4	Compact Sets and the Heine-Borel Theorem	61

	5	Functions	67
	5.1	Limits and Continuity	67
	5.2	Discontinuous Functions	72
	5.3	Properties of Continuous Functions	77
	5.4	Compactness and Extreme Values	80
	5.5	Uniform Continuity	82
	5.6	The Intermediate Value Theorem	88
	5.7	Monotone Functions	90

	6	Differentiation and Integration	94
	6.1	Differentiable Functions	94
	6.2	The Mean Value Theorem	99
	6.3	Riemann Integration	103
	6.4	The Fundamental Theorem of Calculus	109

	7	Norms and Inner Products	113
	7.1	Normed Vector Spaces	113
	7.2	Topology in Normed Spaces	117
	7.3	Finite-Dimensional Normed Spaces	120
	7.4	Inner Product Spaces	124
	7.5	Finite Orthonormal Sets	128
	7.6	Fourier Series	132
	7.7	Orthogonal Expansions and Hilbert Spaces	136

	8	Limits of Functions	142
	8.1	Limits of Functions	142
	8.2	Uniform Convergence and Continuity	147
	8.3	Uniform Convergence and Integration	150
	8.4	Series of Functions	154
	8.5	Power Series	161
	8.6	Compactness and Subsets of C(K)	168

	9	Metric Spaces	175
	9.1	Definitions and Examples	175
	9.2	Compact Metric Spaces	180
	9.3	Complete Metric Spaces	183





Part B		Applications
	10	Approximation by Polynomials	189
	10.1	Taylor Series	189
	10.2	How Not to Approximate a Function	198
	10.3	Bernstein's Proof of the Weierstrass Theorem	201
	10.4	Accuracy of Approximation	204
	10.5	Existence of Best Approximations	207
	10.6	Characterizing Best Approximations	211
	10.7	Expansions Using Chebychev Polynomials	217
	10.8	Splines	223
	10.9	Uniform Approximation by Splines	231
	10.10	The Stone--Weierstrass Theorem	235

	11	Discrete Dynamical Systems	240
	11.1	Fixed Points and the Contraction Principle	240
	11.2	Newton's Method	252
	11.3	Orbits of a Dynamical System	257
	11.4	Periodic Points	262
	11.5	Chaotic Systems	269
	11.6	Topological Conjugacy	277
	11.7	Iterated Function Systems and Fractals	285

	12	Differential Equations	293
	12.1	Integral Equations and Contractions	293
	12.2	Calculus of Vector-Valued Functions	297
	12.3	Differential Equations and Fixed Points	300
	12.4	Solutions of Differential Equations	304
	12.5	Local Solutions	309
	12.6	Linear Differential Equations	316
	12.7	Perturbation and Stability of DEs	320
	12.8	Existence without Uniqueness	324

	13	Fourier Series and Physics	328
	13.1	The Steady-State Heat Equation	328
	13.2	Formal Solution	332
	13.3	Convergence in the Open Disk	334
	13.4	The Poisson Formula	337
	13.5	Poisson's Theorem	341
	13.6	The Maximum Principle	345
	13.7	The Vibrating String (Formal Solution)	347
	13.8	The Vibrating String (Rigourous Solution	353
	13.9	Appendix: The Complex Exponential	356

	14	Fourier Series and Approximation	360
	14.1	The Riemann--Lebesgue Lemma	360
	14.2	Pointwise Convergence of Fourier Series	364
	14.3	Gibbs's Phenomenon	372
	14.4	Cesaro Summation of Fourier Series	376
	14.5	Least Squares Approximations	383
	14.6	The Isoperimetric Problem	387
	14.7	Best Approximation by Trig Polynomials	390
	14.8	Connections with Polynomial Approximation	393
	14.9	Jackson's Theorem and Bernstein's Theorem	397

	15	Wavelets	406
	15.1	Introduction	406
	15.2	The Haar Wavelet	408
	15.3	Multiresolution Analysis	412
	15.4	Recovering the Wavelet	416
	15.5	Daubechies Wavelets	420
	15.6	Existence of the Daubechies Wavelets	426
	15.7	Approximations Using Wavelets	429
	15.8	The Franklin Wavelet	433
	15.9	Riesz Multiresolution Analysis	440

	16	Convexity and Optimization	449
	16.1	Convex Sets	449
	16.2	Relative Interior	455
	16.3	Separation Theorems	460
	16.4	Extreme Points	464
	16.5	Convex Functions in One Dimension	467
	16.6	Convex Functions in Higher Dimensions	473
	16.7	Subdifferentials and Directional Derivatives	477
	16.8	Tangent and Normal Cones	487
	16.9	Constrained Minimization	491
	16.10	The Minimax Theorem	498


		References	505

		Index	507

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Real Analysis and Applications

Table of Contents

Part A

Analysis

Part B

Applications