Edoardo Provenzi - From Euclidean to Hilbert Spaces

1 Chapter 2 Table 2.1. Different normalizations of Fourier bases and relative formulasTable 2.2. Fourier pairs and translationTable 2.3. Fourier pairs relative to convolutionTable 2.4. Fourier pair for the convolution between a signal z and the unit puls...Table 2.5. Fourier pairs for 2D shifts

2 Chapter 5Table 5.1. Analogies between a finite-dimensional Euclidean space and an infinit...

3 Chapter 6Table 6.1. Properties of the Fourier transform on S(ℝ)Table 6.2. Properties of the Fourier transform on S(ℝn)

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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

Edoardo Provenzi

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:

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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

This book provides an introduction to the key theoretical concepts associated with Hilbert spaces and with operators defined over these spaces.

Our decision to dedicate a whole book to the subject of Hilbert spaces stems from a simple observation: of all the infinite dimensional vector spaces, Hilbert spaces bear the closest resemblance to finite dimensional Euclidean spaces, that is, ℝ nor C n, which provide the framework for classical analysis and linear algebra.

The topological subtleties which come into play when using infinite dimensions mean that certain conditions (which are always verified in finite dimensions) must be posed in order to maintain the validity of known results from Euclidian spaces. For Hilbert spaces, one of these topological conditions is completeness, that is, any Cauchy sequence must converge in the space in which it is defined.