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Introduction to Functional Analysis This work is licensed under a Creative Commons “Attribution-NonCommercial-ShareAlike 4.0 Interna- tional” licence. INTRODUCTION TO FUNCTIONAL ANALYSIS VLADIMIR V. KISIL ABSTRACT. This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington and Dr. David Salinger used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more. The notes are available also for download in PDF. The suggested textbooks are [1,9, 11, 12]. The other nice books with many inter- esting problems are [3, 10]. Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them! CONTENTS List of Figures3 Notations and Assumptions4 Integrability conditions4 1. Motivating Example: Fourier Series5 1.1. Fourier series: basic notions5 1.2. The vibrating string8 1.3. Historic: Joseph Fourier 10 2. Basics of Linear Spaces 11 2.1. Banach spaces (basic definitions only) 12 2.2. Hilbert spaces 14 2.3. Subspaces 18 2.4. Linear spans 22 3. Orthogonality 23 3.1. Orthogonal System in Hilbert Space 24 3.2. Bessel’s inequality 25 3.3. The Riesz–Fischer theorem 28 3.4. Construction of Orthonormal Sequences 29 3.5. Orthogonal complements 32 Date: 24th June 2021. 1 2 VLADIMIR V. KISIL 4. Duality of Linear Spaces 33 4.1. Dual space of a normed space 33 4.2. Self-duality of Hilbert space 35 5. Fourier Analysis 36 5.1. Fourier series 36 5.2. Fejer’s´ theorem 38 5.3. Parseval’s formula 43 5.4. Some Application of Fourier Series 44 6. Operators 50 6.1. Linear operators 50 6.2. Orthoprojections 52 6.3. B(H) as a Banach space (and even algebra) 52 6.4. Adjoints 53 6.5. Hermitian, unitary and normal operators 54 7. Spectral Theory 57 7.1. The spectrum of an operator on a Hilbert space 58 7.2. The spectral radius formula 60 7.3. Spectrum of Special Operators 62 8. Compactness 62 8.1. Compact operators 63 8.2. Hilbert–Schmidt operators 66 9. The spectral theorem for compact normal operators 70 9.1. Spectrum of normal operators 70 9.2. Compact normal operators 72 10. Applications to integral equations 74 11. Banach and Normed Spaces 81 11.1. Normed spaces 81 11.2. Bounded linear operators 85 11.3. Dual Spaces 86 11.4. Hahn–Banach Theorem 87 11.5. C(X) Spaces 90 12. Measure Theory 90 12.1. Basic Measure Theory 90 12.2. Extension of Measures 93 12.3. Complex-Valued Measures and Charges 98 12.4. Constructing Measures, Products 100 13. Integration 101 13.1. Measurable functions 101 13.2. Lebsgue Integral 104 13.3. Properties of the Lebesgue Integral 109 13.4. Integration on Product Measures 114 13.5. Absolute Continuity of Measures 117 INTRODUCTION TO FUNCTIONAL ANALYSIS 3 14. Functional Spaces 118 14.1. Integrable Functions 118 14.2. Dense Subspaces in Lp 124 14.3. Continuous functions 127 14.4. Riesz Representation Theorem 130 15. Fourier Transform 134 15.1. Convolutions on Commutative Groups 134 15.2. Characters of Commutative Groups 137 15.3. Fourier Transform on Commutative Groups 140 15.4. The Schwartz space of smooth rapidly decreasing functions 141 15.5. Fourier Integral 142 Appendix A. Tutorial Problems 146 A.1. Tutorial problems I 146 A.2. Tutorial problems II 147 A.3. Tutorial Problems III 148 A.4. Tutorial Problems IV 148 A.5. Tutorial Problems V 149 A.6. Tutorial Problems VI 150 A.7. Tutorial Problems VII 151 Appendix B. Solutions of Tutorial Problems 153 Appendix C. Course in the Nutshell 154 C.1. Some useful results and formulae (1) 154 C.2. Some useful results and formulae (2) 155 Appendix D. Supplementary Sections 158 D.1. Reminder from Complex Analysis 158 References 159 Index 160 LIST OF FIGURES 1 Triangle inequality 13 2 Different unit balls 15 3 To the parallelogram identity. 17 4 Jump function as a limit of continuous functions 19 5 The Pythagoras’ theorem 23 6 Best approximation from a subspace 26 7 Best approximation by three trigonometric polynomials 27 8 Legendre and Chebyshev polynomials 31 9 A modification of continuous function to periodic 37 10 The Fejer´ kernel 40 4 VLADIMIR V. KISIL 11 The dynamics of a heat equation 45 12 Appearance of dissonance 47 13 Different musical instruments 48 14 Fourier series for different musical instruments 49 15 Two frequencies separated in time 49 16 Distance between scales of orthonormal vectors 65 17 The /3 argument to estimate jf(x)- f(y)j. 65 NOTATIONS AND ASSUMPTIONS Z+, R+ denotes non-negative integers and reals. x, y, z,... denotes vectors. λ, µ, ν,... denotes scalars. <z, =z stand for real and imaginary parts of a complex number z. Integrability conditions. In this course, the functions we consider will be real or complex valued functions defined on the real line which are locally Riemann integ- rable. This means that they are Riemann integrable on any finite closed interval [a, b]. (A complex valued function is Riemann integrable iff its real and imagin- ary parts are Riemann-integrable.) In practice, we shall be dealing mainly with bounded functions that have only a finite number of points of discontinuity in any finite interval. We can relax the boundedness condition to allow improper Riemann integrals, but we then require the integral of the absolute value of the function to converge. We mention this right at the start to get it out of the way. There are many fascin- ating subtleties connected with Fourier analysis, but those connected with technical aspects of integration theory are beyond the scope of the course. It turns out that one needs a “better” integral than the Riemann integral: the Lebesgue integral, and I commend the module, Linear Analysis 1, which includes an introduction to that topic which is available to MM students (or you could look it up in Real and Com- plex Analysis by Walter Rudin). Once one has the Lebesgue integral, one can start thinking about the different classes of functions to which Fourier analysis applies: the modern theory (not available to Fourier himself) can even go beyond functions and deal with generalized functions (distributions) such as the Dirac delta function which may be familiar to some of you from quantum theory. From now on, when we say “function”, we shall assume the conditions of the first paragraph, unless anything is stated to the contrary. INTRODUCTION TO FUNCTIONAL ANALYSIS 5 1. MOTIVATING EXAMPLE:FOURIER SERIES 1.1. Fourier series: basic notions. Before proceed with an abstract theory we con- sider a motivating example: Fourier series. 1.1.1. 2π-periodic functions. In this part of the course we deal with functions (as above) that are periodic. We say a function f : R C is periodic with period T > 0 if f(x + T) = f(x) for ! all x R. For example, sin x, cos x, eix(= cos x + i sin x) are periodic with period 2 2π. For k R n f0g, sin kx, cos kx, and eikx are periodic with period 2π/jkj. Constant 2 functions are periodic with period T, for any T > 0. We shall specialize to periodic functions with period 2π: we call them 2π-periodic functions, for short. Note that cos nx, sin nx and einx are 2π-periodic for n Z. (Of course these are also 2π/jnj- 2 periodic.) Any half-open interval of length T is a fundamental domain of a periodic function f of period T. Once you know the values of f on the fundamental domain, you know them everywhere, because any point x in R can be written uniquely as x = w + nT where n Z and w is in the fundamental domain. Thus f(x) = f(w+(n-1)T +T) = 2 = f(w + T) = f(w). ··· For 2π-periodic functions, we shall usually take the fundamental domain to be ]- π, π]. By abuse of language, we shall sometimes refer to [-π, π] as the funda- mental domain. We then have to be aware that f(π) = f(-π). b ikx 1.1.2. Integrating the complex exponential function. We shall need to calculate a e dx, for k R. Note first that when k = 0, the integrand is the constant function 1, so the 2 b ikx b R result is b-a. For non-zero k, a e dx = a(cos kx+i sin kx) dx = (1=k)[(sin kx- i cos kx)]b = (1=ik)[(cos kx + i sin kx)]b = (1=ik)[eikx]b = (1=ik)(eikb - eika). Note a R a R a that this is exactly the result you would have got by treating i as a real constant and using the usual formula for integrating eax. Note also that the cases k = 0 and k = 0 have to be treated separately: this is typical. 6 Definition 1.1. Let f : R C be a 2π-periodic function which is Riemann ! integrable on [-π, π]. For each n Z we define the Fourier coefficient f^(n) by 2 π 1 f^(n) = f(x)e-inx dx . 2π -Zπ Remark 1.2. (i) f^(n) is a complex number whose modulus is the amplitude and whose argument is the phase (of that component of the original function).
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