Functional analysis : entering Hilbert space

The present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i. e. , the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles: Set Theory, Topo? logical Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S. L. SoBOLEV and L. ScmVARTZ. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathe? maticians, both pure and applied. The reader may pass e. g. from Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integ-ration of the Equation of Evolution). Such materials as 'Weak Topologies and Duality in Locally Convex Spaces' and 'Nuclear Spaces' are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators. Contents......Page 10 Preface......Page 8 Preliminary Notions......Page 12 1. Basic Elements of Metric Topology......Page 16 1.1 Metric spaces......Page 16 1.2 The topology of a metric space......Page 20 1.3 Completeness of metric spaces......Page 22 1.4 Normed vector spaces......Page 26 1.5 Bounded linear operators......Page 29 2. New Types of Function Spaces......Page 34 2.1 Completion of metric spaces and normed vector spaces......Page 34 2.2 The Weierstrass Approximation Theorem......Page 39 2.3 Important inequalities for p-norms in spaces of continuous functions......Page 43 2.4 Construction of Lp-spaces......Page 47 2.4.1 The Lp-spaces and some basic inequalities......Page 47 2.4.2 Lebesgue measurable subsets in R......Page 50 2.4.3 Smooth functions with compact support......Page 53 2.4.4 Riemann integrable functions......Page 54 2.5 The sequence spaces lP......Page 56 3. Theory of Hilbert Spaces......Page 60 3.1 Inner product spaces......Page 60 3.2 Hilbert spaces......Page 65 3.3 Basis in a normed vector space and separability......Page 66 3.3.1 Infinite series in normed vector spaces......Page 66 3.3.2 Separability of a normed vector space......Page 67 3.4 Basis in a separable Hilbert space......Page 69 3.5 Orthogonal projection and complement......Page 77 3.6 Weak convergence......Page 82 4. Operators on Hilbert Spaces......Page 86 4.1 The adjoint of a bounded linear operator......Page 86 4.2 Compact operators......Page 93 5. Spectral Theory......Page 100 5.1 The spectrum and the resolvent......Page 100 5.2 Spectral theorem for compact self-adjoint operators......Page 104 Exercises......Page 112 Bibliography......Page 140 List of Symbols......Page 142 Index......Page 144

this Book Presents Basic Elements Of The Theory Of Hilbert Spaces And Operators On Hilbert Spaces, Culminating In A Proof Of The Spectral Theorem For Compact, Self-adjoint Operators On Separable Hilbert Spaces. It Exhibits A Construction Of The Space Of P[superscript Th] Power Lebesgue Integrable Functions By A Completion Procedure With Respect To A Suitable Norm In A Space Of Continuous Functions, Including Proofs Of The Basic Inequalities Of Holder And Minkowski. The L[superscript P]-spaces Thereby Emerges In Direct Analogy With A Construction Of The Real Numbers From The Rational Numbers. This Allows Grasping The Main Ideas More Rapidly. Other Important Banach Spaces Arising From Function Spaces And Sequence Spaces Are Also Treated.

This book presents basic elements of the theory of Hilbert spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, self-adjoint operators on separable Hilbert spaces. It exhibits a construction of the space of pth power Lebesgue integrable functions by a completion procedure with respect to a suitable norm in a space of continuous functions, including proofs of the basic inequalities of Hölder and Minkowski. The Lp-spaces thereby emerges in direct analogy with a construction of the real numbers from the rational numbers. This allows grasping the main ideas more rapidly. Other important Banach spaces arising from function spaces and sequence spaces are also treated. This book presents basic elements of the theory of Hilbert-spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, self-adjoint operators on separable Hilbert spaces.--[book cover]