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AMATH 731: Applied Functional Analysis Lecture Notes

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AMATH 731: Applied Functional Analysis Lecture Notes Sumeet Khatri November 24, 2014 Table of Contents List of Tables ................................................... v List of Theorems ................................................ ix List of Definitions ................................................ xii Preface ....................................................... xiii 1 Review of Real Analysis .......................................... 1 1.1 Convergence and Cauchy Sequences...............................1 1.2 Convergence of Sequences and Cauchy Sequences.......................1 2 Measure Theory ............................................... 2 2.1 The Concept of Measurability...................................3 2.1.1 Simple Functions...................................... 10 2.2 Elementary Properties of Measures................................ 11 2.2.1 Arithmetic in [0, ] .................................... 12 1 2.3 Integration of Positive Functions.................................. 13 2.4 Integration of Complex Functions................................. 14 2.5 Sets of Measure Zero......................................... 14 2.6 Positive Borel Measures....................................... 14 2.6.1 Vector Spaces and Topological Preliminaries...................... 14 2.6.2 The Riesz Representation Theorem........................... 14 2.6.3 Regularity Properties of Borel Measures........................ 14 2.6.4 Lesbesgue Measure..................................... 14 2.6.5 Continuity Properties of Measurable Functions.................... 14 3 Metric Spaces ................................................ 15 3.1 Definition and Examples....................................... 17 3.2 Covergence, Cauchy Sequence, Completeness......................... 19 3.3 The Topology of Metric Spaces................................... 22 3.3.1 Continuity........................................... 26 3.3.2 Equicontinuity........................................ 28 3.3.3 Appendix: Topological Spaces............................... 29 3.4 Equivalent Metrics.......................................... 30 i Chapter 0: TABLE OF CONTENTS 0.0: TABLE OF CONTENTS 3.5 Examples of Complete Metric Spaces............................... 32 3.6 Completion of Metric Spaces.................................... 39 3.7 Lp Spaces................................................ 44 3.8 Appendix: Additional Topics.................................... 45 3.8.1 Pseudomerics......................................... 45 3.8.2 A Metric Space for Sets................................... 46 4 The Contraction Mapping Theorem .................................. 50 4.1 The Theorem.............................................. 50 4.2 Application to Linear Equations.................................. 54 4.3 Application to Ordinary Differential Equations......................... 54 4.3.1 Picard’s Method of Successive Approximations.................... 59 4.4 Application to Integral Equations................................. 61 5 Normed Linear Spaces and Banach Spaces ............................. 62 5.1 Quick Review of Vector Spaces................................... 63 5.2 Norms and Normed Spaces; Banach Spaces........................... 65 5.2.1 Sequences and Convergence; Bases........................... 70 5.2.2 Completeness......................................... 73 5.2.3 Compactness......................................... 74 5.2.4 Equivalent Norms...................................... 77 5.2.5 Convexity........................................... 80 5.3 The Schauder Fixed Point Theorem................................ 81 5.3.1 Application to Ordinary Differential Equations.................... 85 5.4 Linear Operators........................................... 87 5.5 Bounded and Continuous Linear Operators........................... 91 5.5.1 Inverse of Linear Operators................................ 100 5.5.2 Linear Functionals...................................... 103 5.6 Representing Linear Operators and Functionals on Finite-Dimensional Spaces...... 105 5.7 Normed Spaces of Operators.................................... 106 5.7.1 Convergence of Sequences of Operators and Functionals.............. 107 5.7.2 The Dual Space....................................... 109 5.7.3 Series Expansions of Bounded Linear Operators.................... 111 ii Chapter 0: TABLE OF CONTENTS 0.0: TABLE OF CONTENTS 5.7.4 Application: The Neumann Series............................ 113 5.8 The Hahn-Banach Theorem..................................... 117 5.8.1 Application to Bounded Linear Functions on C[a, b] ................. 119 5.8.2 The Adjoint Operator.................................... 121 5.9 The Fréchet Derivative........................................ 124 5.9.1 The Generalised Mean Value Theorem......................... 130 5.9.2 Application: The Newton-Kantorovich Method.................... 135 5.9.3 Application: Stability of Dynamical Systems...................... 141 6 Inner Product Spaces and Hilbert Spaces ..............................143 6.1 Definition and Examples....................................... 143 6.2 Properties of Inner Product and Hilbert Spaces......................... 151 6.2.1 Completion.......................................... 151 6.2.2 Orthogonality......................................... 151 6.2.3 Orthonormal Sets and Sequences............................ 158 6.2.4 Series Related to Orthonormal Sequences and Sets.................. 172 6.3 Total Orthonormal Sets and Sequences.............................. 175 6.3.1 Legendre, Laguerre, and Hermite Polynomials..................... 180 6.4 Representation of Functionals................................... 180 6.5 The Hilbert Adjoint Operator.................................... 185 6.6 Self-Adjoint, Unitary and Normal Operators........................... 191 6.6.1 Application: The Fourier Transform........................... 196 6.6.2 Application: Quantum Mechanics............................ 200 6.7 Compact Operators.......................................... 200 6.8 Closed Linear Operators....................................... 209 7 Spectral Theory ...............................................213 7.1 Finite-Dimensional Normed Spaces................................ 213 7.2 General Normed Spaces....................................... 217 7.3 Bounded Linear Operators on Normed Spaces......................... 221 7.4 Compact Linear Operators on Normed Spaces......................... 225 7.4.1 Operator Equations Involving Compact Linear Operators.............. 228 7.5 Bounded Self-Adjoint Linear Operators on Hilbert Spaces.................. 228 iii Chapter 0: 0.0: 7.5.1 Compact Self-Adjoint Operators; The Spectral Theorem............... 233 7.5.2 Positive Operators...................................... 236 7.6 Projection Operators......................................... 239 7.7 Spectral Family............................................ 243 7.7.1 Bounded Self-Adjoint Linear Operators......................... 244 7.8 Spectral Decomposition of Bounded Self-Adjoint Linear Operators............. 244 7.8.1 The Spectral Theorem for Continuous Functions................... 244 7.9 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator....... 244 7.10 Sturm-Lioville Problems....................................... 244 7.11 Appendix: Banach Algebras..................................... 244 7.12 Appendix: C ∗-Algebras........................................ 244 8 Sobolev Spaces ...............................................245 iv List of Tables v List of Theorems 2.1.1 Theorem................................................4 2.1.2 Theorem................................................5 2.1.3 Theorem................................................6 2.1.4 Theorem................................................7 2.1.5 Theorem................................................9 2.1.6 Theorem................................................9 2.1.7 Theorem................................................ 10 2.2.1 Theorem (Important Properties of Measures).......................... 11 3.2.1 Theorem (Convergent Sequences)................................ 22 3.3.1 Theorem................................................ 25 3.3.2 Theorem................................................ 26 3.3.3 Theorem................................................ 27 3.3.4 Theorem................................................ 28 3.3.5 Theorem................................................ 28 3.3.6 Theorem................................................ 28 3.3.7 Theorem (Arzela-Ascoli)...................................... 29 3.5.1 Theorem (Complete Subspace).................................. 32 ` ` 3.5.2 Theorem (Completeness of R and C ).............................. 32 3.5.3 Theorem (Completeness of ` ).................................. 34 1 3.5.4 Theorem (Completeness of (`c, , d ))............................. 35 1 1 3.5.5 Theorem (Completeness of (`p, dp))............................... 35 3.5.6 Theorem................................................ 36 3.5.7 Theorem (Uniform Convergence)................................. 37 3.6.1 Theorem (Weierstrass Approximation Theorem)........................ 40 3.6.2 Theorem (Completion)....................................... 41 3.7.1 Theorem (Riesz-Fischer)...................................... 44 4.1.1 Theorem (Contraction Mapping/Banach Fixed Point Theorem)............... 51 4.1.2 Theorem (Contraction on a Ball)................................. 53 4.3.1 Theorem (Picard’s Existence and
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