Edoardo Provenzi - From Euclidean to Hilbert Spaces

1 Cover

2 Dedication To my mentors, Sissa Abbati and Renzo Cirelli, who taught me the importance of rigor in mathematics, and to Brunella, Paola, Clara and Tommo, whose passion for their work has both helped and brought joy to many

3 Title Page

4 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Edoardo Provenzi to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021937006 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-682-1

5 Preface

6 1 Inner Product Spaces (Pre-Hilbert) 1.1. Real and complex inner products 1.2. The norm associated with an inner product and normed vector spaces 1.3. Orthogonal and orthonormal families in inner product spaces 1.4. Generalized Pythagorean theorem 1.5. Orthogonality and linear independence 1.6. Orthogonal projection in inner product spaces 1.7. Existence of an orthonormal basis: the Gram-Schmidt process 1.8. Fundamental properties of orthonormal and orthogonal bases 1.9. Summary

7 2 The Discrete Fourier Transform and its Applications to Signal and Image Processing 2.1. The space ℓ 2(ℤ N) and its canonical basis 2.2. The orthonormal Fourier basis of ℓ 2(ℤ N) 2.3. The orthogonal Fourier basis of ℓ 2(ℤ N) 2.4. Fourier coefficients and the discrete Fourier transform 2.5. Matrix interpretation of the DFT and the IDFT 2.6. The Fourier transform in signal processing 2.7. Properties of the DFT 2.8. The DFT and stationary operators 2.9. The two-dimensional discrete Fourier transform (2D DFT) 2.10. Summary

8 3 Lebesgue’s Measure and Integration Theory 3.1. Riemann versus Lebesgue 3.2. σ -algebra, measurable space, measures and measured spaces 3.3. Measurable functions and almost-everywhere properties (a.e) 3.4. Integrable functions and Lebesgue integrals 3.5. Characterization of the Lebesgue measure on ℝ and sets with a null Lebesgue measure 3.6. Three theorems for limit operations in integration theory 3.7. Summary

9 4 Banach Spaces and Hilbert Spaces 4.1. Metric topology of inner product spaces 4.2. Continuity of fundamental operations in inner product spaces 4.3. Cauchy sequences and completeness: Banach and Hilbert 4.4. Remarkable examples of Banach and Hilbert spaces 4.5. Summary

10 5 The Geometric Structure of Hilbert Spaces 5.1. The orthogonal complement in a Hilbert space and its properties 5.2. Projection onto closed convex sets: theorem and consequences 5.3. Polar and bipolar subsets of a Hilbert space 5.4. The (orthogonal) projection theorem in a Hilbert space 5.5. Orthonormal systems and Hilbert bases 5.6. The Fourier Hilbert basis in L 2 5.7. Summary

11 6 Bounded Linear Operators in Hilbert Spaces 6.1. Fundamental properties of bounded linear operators between normed vector spaces 6.2. The operator norm, convergence of operator sequences and Banach algebras 6.3. Invertibility of linear operators 6.4. The dual of a Hilbert space and the Riesz representation theorem 6.5. Bilinear forms, sesquilinear forms and associated quadratic forms 6.6. The adjoint operator: presentation and properties 6.7. Orthogonal projection operators in a Hilbert space 6.8. Isometric and unitary operators 6.9. The Fourier transform on S (ℝ n), L 1(ℝ n) and L 2(ℝ n) 6.10. The Nyquist-Shannon sampling theorem 6.11. Application of the Fourier transform to solve ordinary and partial differential equations 6.12. Summary

12 Appendix 1: Quotient Space

13 Appendix 2: The Transpose (or Dual) of a Linear Operator

14 Appendix 3: Uniform, Strong and Weak Convergence

15 References

16 Index

17 End User License Agreement

1 Cover

2 Table of Contents

3 Dedication To my mentors, Sissa Abbati and Renzo Cirelli, who taught me the importance of rigor in mathematics, and to Brunella, Paola, Clara and Tommo, whose passion for their work has both helped and brought joy to many

4 Title Page

5 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Edoardo Provenzi to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021937006 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-682-1

6 Preface

7 Begin Reading

8 Index

9 End User License Agreement

1 Chapter 1 Figure 1.1. Parallelogram law in ℝ2: The sum of the squares of the two diagonal ... Figure 1.2. Orthogonal projection and diagonal projections and of a vector in... Figure 1.3. Visualization of property 2 in ℝ2. For a color version of this figur... Figure 1.4. Orthogonal projection p of a vector in ℝ3 onto the plane produced by... Figure 1.5. Illustration of the second step in the Gram-Schmidt orthonormalizati...

2 Chapter 2 Figure 2.1. Hm for m = 1, N = 16 Figure 2.2. Hm for m = 2, N = 16 Figure 2.3. Filtering approach in the Fourier domainFigure 2.4. Difference between the sine functions representing the spectrum valu...Figure 2.5. The two coordinates of a pixel, n1 n2, in a digital image (image sou...Figure 2.6. a) Original image of Panko; b) image after Laplacian filter; c) imag...Figure 2.7. Left column: original images. Right column: centered amplitude spect...Figure 2.8. Two-dimensional Gaussian images with a standard deviation of (left -...Figure 2.9. Blurred image of Lena obtained by multiplying DFTs and Gaussians wit...Figure 2.10. Blurring filter/low-pass filter in the frequency domain

3 Chapter 3Figure 3.1. Riemann and Lebesgue integration. For a color version of this figure...

4 Chapter 5Figure 5.1. Two-dimensional geometric visualization of the property verified by ...Figure 5.2. Gibbs phenomenon for the rectangular pulse function (courtesy of Éri...

5 Chapter 6Figure 6.1. The line of equation (shown in blue) is the best approximation of t...