This book introduces functional analysis at an elementary level without assuming any background in real analysis, for example on metric spaces or Lebesgue integration. It focuses on concepts and methods relevant in applied contexts such as variational methods on Hilbert spaces, Neumann series, eigenvalue expansions for compact self-adjoint operators, weak differentiation and Sobolev spaces on intervals, and model applications to differential and integral equations. Beyond that, the final chapters on the uniform boundedness theorem, the open mapping theorem and the Hahn-Banach theorem provide a stepping-stone to more advanced texts. The exposition is clear and rigorous, featuring full and detailed proofs. Many examples illustrate the new notions and results. Each chapter concludes with a large collection of exercises, some of which are referred to in the margin of the text, tailor-made in order to guide the student digesting the new material. Optional sections and chapters supplement the mandatory parts and allow for modular teaching spanning from basic to honors track level.
Preface xiii
Chapter 1 Inner Product Spaces 1 (14)
§1.1 Inner Products 3 (3)
§1.2 Orthogonality 6 (4)
§1.3 The Trigonometric System 10 (5)
Exercises 11 (4)
Chapter 2 Normed Spaces 15 (22)
§2.1 CThe Cauchy--Schwarz Inequality 15 (3)
and the Space l2
§2.2 Norms 18 (3)
§2.3 Bounded Linear Mappings 21 (2)
§2.4 Basic Examples 23 (5)
§2.5 The P-Spaces (1 ≤ p < 28 (9)
∞)
Exercises 31 (6)
Chapter 3 Distance and Approximation 37 (18)
§3.1 Metric Spaces 37 (2)
§3.2 Convergence 39 (2)
§3.3 Uniform, Pointwise and (Square) 41 (6)
Mean Convergence
§3.4 The Closure of a Subset 47 (8)
Exercises 50 (5)
Chapter 4 Continuity and Compactness 55 (24)
§4.1 Open and Closed Sets 55 (3)
§4.2 Continuity 58 (6)
§4.3 Sequential Compactness 64 (2)
§4.4 Equivalence of Norms 66 (5)
§4.5 *Separability and General 71 (8)
Compactness
Exercises 74 (5)
Chapter 5 Banach Spaces 79 (14)
§5.1 Cauchy Sequences and 79 (2)
Completeness
§5.2 Hilbert Spaces 81 (3)
§5.3 Banach Spaces 84 (2)
§5.4 Series in Banach Spaces 86 (7)
Exercises 88 (5)
Chapter 6 *The Contraction Principle 93 (14)
§6.1 Banach's Contraction Principle 94 (1)
§6.2 Application: Ordinary 95 (3)
Differential Equations
§6.3 Application: Google's PageRank 98 (2)
§6.4 Application: The Inverse 100 (7)
Mapping Theorem
Exercises 104 (3)
Chapter 7 The Lebesgue Spaces 107 (22)
§7.1 The Lebesgue Measure 110 (3)
§7.2 The Lebesgue Integral and the 113 (2)
Space L1(X)
§7.3 Null Sets 115 (3)
§7.4 The Dominated Convergence 118 (3)
Theorem
§7.5 The Spaces LP(X) with 1 ≤ p 121 (8)
< ∞
Advice for the Reader 125 (1)
Exercises 126 (3)
Chapter 8 Hilbert Space Fundamentals 129 (18)
§8.1 Best Approximations 129 (4)
§8.2 Orthogonal Projections 133 (2)
§8.3 The Riesz--Frechet Theorem 135 (2)
§8.4 Orthogonal Series and Abstract 137 (10)
Fourier Expansions
Exercises 141 (6)
Chapter 9 Approximation Theory and Fourier 147 (30)
Analysis
§9.1 Lebesgue's Proof of 149 (2)
Weierstrass' Theorem
§9.2 Truncation 151 (5)
§9.3 Classical Fourier Series 156 (5)
§9.4 Fourier Coefficients of 161 (1)
L1-Functions
§9.5 The Riemann--Lebesgue Lemma 162 (2)
§9.6 *The Strong Convergence Lemma 164 (4)
and Fejer's Theorem
§9.7 *Extension of a Bounded Linear 168 (9)
Mapping
Exercises 172 (5)
Chapter 10 Sobolev Spaces and the Poisson 177 (16)
Problem
§10.1 Weak Derivatives 177 (2)
§10.2 The Fundamental Theorem of 179 (3)
Calculus
§10.3 Sobolev Spaces 182 (2)
§10.4 The Variational Method for the 184 (3)
Poisson Problem
§10.5 *Poisson's Problem in Higher 187 (6)
Dimensions
Exercises 188 (5)
Chapter 11 Operator Theory I 193 (18)
§11.1 Integral Operators and 193 (3)
Fubini's Theorem
§11.2. The Dirichlet Laplacian and 196 (3)
Hilbert--Schmidt Operators
§11.3 Approximation of Operators 199 (3)
§11.4 The Neumann Series 202 (9)
Exercises 205 (6)
Chapter 12 Operator Theory II 211 (20)
§12.1 Compact Operators 211 (5)
§12.2 Adjoints of Hilbert Space 216 (3)
Operators
§12.3 *The Lax--Milgram Theorem 219 (2)
§12.4 *Abstract Hilbert--Schmidt 221 (10)
Operators
Exercises 226 (5)
Chapter 13 Spectral Theory of Compact 231 (16)
Self-Adjoint Operators
§13.1 Approximate Eigenvalues 231 (3)
§13.2 Self-Adjoint Operators 234 (2)
§13.3 The Spectral Theorem 236 (4)
§13.4 *The General Spectral Theorem 240 (7)
Exercises 241 (6)
Chapter 14 Applications of the Spectral 247 (14)
Theorem
§14.1 The Dirichlet Laplacian 247 (2)
§14.2 The Schrodinger Operator 249 (3)
§14.3 An Evolution Equation 252 (2)
§14.4 *The Norm of the Integration 254 (2)
Operator
§14.5 *The Best Constant in the 256 (5)
Poincare Inequality
Exercises 257 (4)
Chapter 15 Baire's Theorem and Its 261 (16)
Consequences
§15.1 Baire's Theorem 261 (2)
§15.2 The Uniform Boundedness 263 (3)
Principle
§15.3 Nonconvergence of Fourier 266 (1)
Series
§15.4 The Open Mapping Theorem 267 (4)
§15.5 Applications with a Look Back 271 (6)
Exercises 274 (3)
Chapter 16 Duality and the Hahn--Banach 277 (28)
Theorem
§16.1 Extending Linear Functionals 278 (6)
§16.2 Elementary Duality Theory 284 (5)
§16.3 Identification of Dual Spaces 289 (6)
§16.4 *The Riesz Representation 295 (10)
Theorem
Exercises 299 (6)
Historical Remarks 305 (50)
Appendix A Background 311 (22)
§A.1 Sequences and Subsequences 311 (1)
§A.2 Equivalence Relations 312 (2)
§A.3 Ordered Sets 314 (2)
§A.4 Countable and Uncountable Sets 316 (1)
§A.5 Real Numbers 316 (5)
§A.6 Complex Numbers 321 (1)
§A.7 Linear Algebra 322 (7)
§A.8 Set-theoretic Notions 329 (4)
Appendix B The Completion of a Metric 333 (6)
Space
§B.1 Uniqueness of a Completion 334 (1)
§B.2 Existence of a Completion 335 (2)
§B.3 The Completion of a Normed 337 (1)
Space
Exercises 338 (1)
Appendix C Bernstein's Proof of 339 (4)
Weierstrass' Theorem
Appendix D Smooth Cutoff Functions 343 (2)
Appendix E Some Topics from Fourier 345 (6)
Analysis
§E.1 Plancherel's Identity 346 (1)
§E.2 The Fourier Inversion Formula 347 (1)
§E.3 The Carlson--Beurling 348 (1)
Inequality
Exercises 349 (2)
Appendix F General Orthonormal Systems 351 (4)
§F.1 Unconditional Convergence 351 (2)
§F.2 Uncountable Orthonormal Bases 353 (2)
Bibliography 355 (4)
Symbol Index 359 (2)
Subject Index 361 (10)
Author Index 371
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