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Companion to Functional Analysis John M. Erdman Portland State

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Companion to Functional Analysis John M. Erdman Portland State Companion to Functional Analysis John M. Erdman Portland State University Version May 23, 2013 c 2012 John M. Erdman E-mail address: [email protected] Contents PREFACE 1 Greek Letters 2 Fraktur Fonts 3 Chapter 1. LINEAR ALGEBRA AND THE SPECTRAL THEOREM5 1.1. Vector Spaces and the Decomposition of Diagonalizable Operators5 1.2. Normal Operators on an Inner Product Space 10 Chapter 2. A VERY BRIEF DIGRESSION ON THE LANGUAGE OF CATEGORIES 19 2.1. Objects and Morphisms 19 2.2. Functors 21 Chapter 3. NORMED LINEAR SPACES 25 3.1. Norms 25 3.2. Bounded Linear Maps 29 3.3. Finite Dimensional Spaces 31 3.4. Quotients of Normed Linear Spaces 31 3.5. Products of Normed Linear Spaces 32 3.6. Coproducts of Normed Linear Spaces 36 Chapter 4. HILBERT SPACES 39 4.1. Definition and Examples 39 4.2. Nets 40 4.3. Unordered Summation 43 4.4. Hilbert space Geometry 45 4.5. Orthonormal Sets and Bases 47 4.6. The Riesz-Fr´echet Theorem 48 4.7. Strong and Weak Topologies on Hilbert Spaces 49 4.8. Completion of an Inner Product Space 51 Chapter 5. HILBERT SPACE OPERATORS 55 5.1. Invertible Linear Maps and Isometries 55 5.2. Operators and their Adjoints 56 5.3. Algebras with Involution 60 5.4. Self-Adjoint Operators 61 5.5. Projections 62 5.6. Normal Operators 64 5.7. Operators of Finite Rank 64 Chapter 6. BANACH SPACES 67 6.1. The Hahn-Banach Theorems 67 6.2. Natural Transformations 69 6.3. Banach Space Duality 70 6.4. The Open Mapping Theorem 75 6.5. The Closed Graph Theorem 77 iii iv CONTENTS 6.6. Projections and Complemented Subspaces 78 6.7. The Principle of Uniform Boundedness 78 Chapter 7. COMPACT OPERATORS 81 7.1. Definition and Elementary Properties 81 7.2. Trace Class Operators 83 7.3. Hilbert-Schmidt Operators 84 Chapter 8. SOME SPECTRAL THEORY 87 8.1. The Spectrum 87 8.2. Spectra of Hilbert Space Operators 91 Chapter 9. TOPOLOGICAL VECTOR SPACES 93 9.1. Balanced Sets and Absorbing Sets 93 9.2. Filters 93 9.3. Compatible Topologies 94 9.4. Quotients 97 9.5. Locally Convex Spaces and Seminorms 98 9.6. Fr´echet Spaces 99 Chapter 10. DISTRIBUTIONS 103 10.1. Inductive Limits 103 10.2. LF -spaces 104 10.3. Distributions 105 10.4. Convolution 108 10.5. Distributional Solutions to Ordinary Differential Equations 111 10.6. The Fourier Transform 112 Chapter 11. THE GELFAND-NAIMARK THEOREM 115 11.1. Maximal Ideals in C(X) 115 11.2. The Character Space 115 11.3. The Gelfand Transform 118 11.4. Unital C∗-algebras 121 11.5. The Gelfand-Naimark Theorem 122 Chapter 12. MULTIPLICATIVE IDENTITIES AND THEIR ALTERNATIVES 125 12.1. Quasi-inverses 125 12.2. Modular Ideals 127 12.3. Unitization of C∗-algebras 128 12.4. Positive Elements in a C∗-Algebra 133 12.5. Approximate Identities 135 Chapter 13. THE GELFAND-NAIMARK-SEGAL CONSTRUCTION 137 13.1. Positive Linear Functionals 137 13.2. Representations 137 13.3. The GNS-Construction and the Third Gelfand-Naimark Theorem 138 Chapter 14. MULTIPLIER ALGEBRAS 141 14.1. Hilbert Modules 141 14.2. Essential Ideals 144 14.3. Compactifications and Unitizations 146 Chapter 15. THE K0-FUNCTOR 149 15.1. Partial Isometries 149 15.2. Equivalence Relations on Projections 150 CONTENTS v 15.3. A Semigroup of Projections 153 15.4. The Grothendieck Construction 155 ∗ 15.5. The K0 Group for Unital C -Algebras 157 15.6. K0(A)|the Nonunital Case 159 15.7. Exactness and Stability Properties of the K0 Functor 160 Bibliography 161 Index 163 PREFACE Paul Halmos famously remarked in his beautiful Hilbert Space Problem Book [22] that \The only way to learn mathematics is to do mathematics." Halmos is certainly not alone in this belief. The current set of notes is an activity-oriented companion to the study of functional analysis. It is intended as a pedagogical companion for the beginner, an introduction to some of the main ideas in functional analysis, a compendium of problems I think are useful in learning the subject, and an annotated reading/reference list. The great majority of the results in beginning functional analysis are straightforward and can be verified by the thoughtful student. Indeed, that is the main point of these notes|to convince the beginner that the subject is accessible. In the material that follows there are numerous indicators that suggest activity on the part of the reader: words such as \proposition", \example", \exercise", and \corollary", if not followed by a proof or a reference to a proof, are invitations to verify the assertions made. Of course, there are a few theorems where, in my opinion, the time and effort which go into proving them is greater than the benefits derived. In such cases, when I have no improvements to offer to the standard proofs, instead of repeating them I give references. These notes were written for a year long course in functional analysis for graduate students at Portland State University. During the year students are asked to choose, in addition to these notes, other sources of information for the course, either printed texts or online documents, which suit their individual learning styles. As a result the material that is intended to be covered during the Fall quarter, the first 6{8 chapters, is relatively complete. After that, when students have found other sources they like, the notes become sketchier. There are of course a number of advantages and disadvantages in consigning a document to electronic life. One advantage is the rapidity with which links implement cross-references. Hunting about in a book for lemma 3.14.23 can be time-consuming (especially when an author engages in the entirely logical but utterly infuriating practice of numbering lemmas, propositions, theorems, corollaries, and so on, separately). A perhaps more substantial advantage is the ability to correct errors, add missing bits, clarify opaque arguments, and remedy infelicities of style in a timely fashion. The correlative disadvantage is that a reader returning to the web page after a short time may find everything (pages, definitions, theorems, sections) numbered differently. (LATEXis an amazing tool.) I will change the date on the title page to inform the reader of the date of the last nontrivial update (that is, one that affects numbers or cross-references). The most serious disadvantage of electronic life is impermanence. In most cases when a web page vanishes so, for all practical purposes, does the information it contains. For this reason (and the fact that I want this material to be freely available to anyone who wants it) I am making use of a \Share Alike" license from Creative Commons. It is my hope that anyone who finds this material useful will correct what is wrong, add what is missing, and improve what is clumsy. For more information on creative commons licenses see http://creativecommons.org/. Concerning the text itself, please send corrections, suggestions, complaints, and all other comments to the author at [email protected] 1 2 PREFACE Greek Letters Upper case Lower case English name (approximate pronunciation) A α Alpha (AL-fuh) B β Beta (BAY-tuh) Γ γ Gamma (GAM-uh) ∆ δ Delta (DEL-tuh) E or " Epsilon (EPP-suh-lon) Z ζ Zeta (ZAY-tuh) H η Eta (AY-tuh) Θ θ Theta (THAY-tuh) I ι Iota (eye-OH-tuh) K κ Kappa (KAP-uh) Λ λ Lambda (LAM-duh) M µ Mu (MYOO) N ν Nu (NOO) Ξ ξ Xi (KSEE) O o Omicron (OHM-ih-kron) Π π Pi (PIE) P ρ Rho (ROH) Σ σ Sigma (SIG-muh) T τ Tau (TAU) Y υ Upsilon (OOP-suh-lon) Φ φ Phi (FEE or FAHY) X χ Chi (KHAY) Ψ Psi (PSEE or PSAHY) Ω ! Omega (oh-MAY-guh) FRAKTUR FONTS 3 Fraktur Fonts In these notes Fraktur fonts are used (most often for families of sets and families of operators). Below are the Roman equivalents for each letter. When writing longhand or presenting material on a blackboard it is usually best to substitute script English letters. Fraktur Fraktur Roman Upper case Lower case Lower Case A a a B b b C c c D d d E e e F f f G g g H h h I i i J j j K k k L l l M m m N n n O o o P p p Q q q R r r S s s T t t U u u V v v W w w X x x Y y y Z z z CHAPTER 1 LINEAR ALGEBRA AND THE SPECTRAL THEOREM The emphasis in beginning analysis courses is on the behavior of individual (real or complex valued) functions. In a functional analysis course the focus is shifted to spaces of such functions and certain classes of mappings between these spaces. It turns out that a concise, perhaps overly simplistic, but certainly not misleading, description of such material is infinite dimensional linear algebra. From an analyst's point of view the greatest triumphs of linear algebra were, for the n n most part, theorems about operators on finite dimensional vector spaces (principally R and C ). However, the vector spaces of scalar-valued mappings defined on various domains are typically infinite dimensional. Whether one sees the process of generalizing the marvelous results of finite dimensional linear algebra to the infinite dimensional realm as a depressingly unpleasant string of abstract complications or as a rich source of astonishing insights, fascinating new questions, and suggestive hints towards applications in other fields depends entirely on one's orientation towards mathematics in general.
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