ISBN-13 (Print): 978-3-319-06727-8 ISBN-13 (Online): 978-3-319-06728-5

Provides a self-contained introduction to functional analysis, assuming only real analysis and linear algebra Presents the essential aspects of metric spaces, Hilbert spaces, Banach spaces and Banach algebras Includes interesting applications of Hilbert spaces such as least squares approximation and inverse problems Prepares the reader for graduate-level mathematical analysis This textbook is an introduction to functional analysis suited to final year undergraduates or beginning graduates. Its various applications of Hilbert spaces, including least squares approximation, inverse problems, and Tikhonov regularization, should appeal not only to mathematicians interested in applications, but also to researchers in related fields. Functional Analysis adopts a self-contained approach to Banach spaces and operator theory that covers the main topics, based upon the classical sequence and function spaces and their operators. It assumes only a minimum of knowledge in elementary linear algebra and real analysis; the latter is redone in the light of metric spaces. It contains more than a thousand worked examples and exercises, which make up the main body of the book. Book review by Prof. Mark Hunacek for the Mathematical Association of America (MAA). Order a copy from Springer. Request an online book review copy. (For Instructors only. Please indicate your university homepage.) Table of Contents Introduction: Preliminaries Part I Metric Spaces Distance: Balls and Open Sets; Closed Sets Convergence and Continuity: Convergence; Continuity Completeness and Separability: Completeness; Uniformly Continuous Maps; Separable Spaces Connectedness: Connected Sets; Components Compactness: Bounded Sets; Totally Bounded Sets; Compact Sets; The Space C(X,Y) Part II Banach and Hilbert Spaces Normed Spaces: Vector Spaces; Norms; Metric and Vector Properties; Complete and Separable Normed Vector Spaces; Series Continuous Linear Maps: Operators; Quotient Spaces; \(R^N\) and Totally Bounded Sets Main Examples: Sequence Spaces; Function Spaces Hilbert Spaces: Inner Products; Least Squares Approximation; Duality; The Adjoint Map; Inverse Problems; Orthonormal Bases Banach Spaces: The Open Mapping Theorem; Compact Operators; The Dual Space; The Adjoint; Pointwise and Weak Convergence Differentiation and Integration: Differentiation; Integration for Vector-Valued Functions; Complex Differentiation and Integration Part III Banach Algebras Banach Algebras: Introduction; Power Series; The Group of Invertible Elements; Analytic Functions Spectral Theory: The Spectral Radius; The Spectrum of an Operator; Spectra of Compact Operators; The Functional Calculus; The Gelfand Transform C*-Algebras: Normal Elements; Normal Operators in B(H); The Spectral Theorem for Compact Normal Operators; Representation Theorems Hints to Selected Problems Glossary of Symbols Further Reading References Index

List of Errata

• Page 18 Examples 2.8 (1) "For \(x=a\),..., that are inside \(B_\epsilon(a)\)" Replace \(B_\epsilon(a)\) with \(]a,b[\) (K. Koifman)
• Page 33 Paragraph before Prop. 3.9 "The following three propositions..." Should be two propositions. (K. Koifman)
• Page 35 Ex.10 Replace "(in \(X=Y=\mathbb{R}\))" with \((X,Y\subseteq\mathbb{R}^2)\).
• Page 50 Line 6 The point \(y\) should be \(x'\).
• Page 50 Examples 4.15 (1) "\(\exists \xi\in]0,1[\)" should be \(\exists \xi\in]a,b[\). (K. Koifman)
• Page 55 Remark 1. The conclusion should be "has dense complement" not "is nowhere dense". (N. Stevenson)
• Page 80 2nd paragraph (mid) There is a missing bracket in \((x_i+\delta,f(x_i+\delta))\). (K. Koifman)
• Page 95 Examples 7.3 (5) There is a missing bracket in the definition of addition of sequences. (K. Koifman)
• Page 103 Example 7.11 1(b) 5th line \(C[0,1]\) should be \(C[a,b]\). (K. Koifman)
• Page 110 Prop.7.21 end of proof, last displayed equation, \(x_{n_1}-x_{n_r}\) should be \(x_{n_1}-x_{n_{r+1}}\), and \(x_{n_{r+1}}-x_{n_r}\) should be \(x_{n_r}-x_{n_{r+1}}\). (K. Koifman)
• Page 120 Prop.8.5 proof, part (iii), \(\|f\|_{L^\infty(B)}\) should be \(\|Tf\|_{L^\infty(B)}\) Part (iv) center formula should have \(dy\) inserted. (K. Koifman)
• Page 125 Ex.8.9(4) proof, replace with "Then, for any other vector \(y\in\tilde{X}\), with \(y_n\to y, y\in X\)". (K. Koifman)
• Page 127 Ex.8.10(19) "kernel" not "kernal".
• Page 128 Proof of Prop. 8.12. "Let \(T\) be a bijective linear map, let \(x,y\in Y\)" (not \(X\)).
• Page 132 Last line: "x+M=0+M" should be on one line.
• Page 133 Ex.8.19(1) There is an extra "xs" at the end of the statement.
• Page 137 Ex.8.25(4) To clarify: Among normed spaces, only in finite dimensions are closed and bounded subsets compact.
• Page 150 In the proof of Prop.9.11, the last inequality is reversed; it follows from (9.2) \((\alpha x+\beta y)^r\le(\alpha x^r+\beta y^r)\) for \(r\ge1\) by putting \(x=a^p, y=b^p, r=q/p\).
• Page 163 Prop.9.21 The statement "\(h_n\to0\) uniformly on \(\mathbb{R}\setminus[-\delta,\delta]\)" should be replaced by "\(h_n\to0\) in \(L^1\) on \(\mathbb{R}\setminus[-\delta,\delta]\)". In the proof, equation (9.7), replace \(0\le h_n(y)<\epsilon\) by \(0\le\int_{\mathbb{R}\setminus[-\delta,\delta]}h_n<\epsilon\).
• Page 182 Last equation, replace \(\langle y_{N,x}\rangle\) with \(\langle y_N, x\rangle\). (K.Koifman)
• Page 254 Example 11.41(7) The assertion holds for Hilbert spaces (since Note 6 is used).
• Page 320 Prop.14.15 Part (ii) should read "finite descent", part (iii) should read "finite ascent and descent".
• Page 329 Prop.14.27 Part (i) The "textbf" should not be there.
• Page 364 Ex.15.24(6) Should be \(T^*Tx=|\lambda|^2x\). Further on, \(e_n\) are the eigenvectors of \(T\) with non-zero eigenvalues \(\lambda_n\), so \(\langle e_n,x\rangle\ne0\) for some \(n\).
• Page 394 Ex. 4.17(1b) First inequality should be \(|(x_2-x_1)y_2+x_1(y_2-y_1)|\le(|y_2||x_1-x_2|+|x_1||y_2-y_1|)\le(|x_1-x_2|+|y_2-y_1|)\) (N. Stevenson)
• Page 398 Ex.9.15(2) Answer should be \(|a_n|^{p'/p}e^{-i\theta_n}\)
• Page 398 Ex.10.15(1) Answer should be \(\frac{1}{14}(10x_0-2y_0+6z_0, -2x_0+13y_0+3z_0, 6x_0+3y_0+5z_0)\) (K. Abela).