From Euclidean To Hilbert Spaces
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Author | : Edoardo Provenzi |
Publisher | : John Wiley & Sons |
Total Pages | : 370 |
Release | : 2021-08-24 |
Genre | : Mathematics |
ISBN | : 1786306824 |
From Euclidian to Hilbert Spaces analyzes the transition from finite dimensional Euclidian spaces to infinite-dimensional Hilbert spaces, a notion that can sometimes be difficult for non-specialists to grasp. The focus is on the parallels and differences between the properties of the finite and infinite dimensions, noting the fundamental importance of coherence between the algebraic and topological structure, which makes Hilbert spaces the infinite-dimensional objects most closely related to Euclidian spaces. The common thread of this book is the Fourier transform, which is examined starting from the discrete Fourier transform (DFT), along with its applications in signal and image processing, passing through the Fourier series and finishing with the use of the Fourier transform to solve differential equations. The geometric structure of Hilbert spaces and the most significant properties of bounded linear operators in these spaces are also covered extensively. The theorems are presented with detailed proofs as well as meticulously explained exercises and solutions, with the aim of illustrating the variety of applications of the theoretical results.
Author | : Philippe Blanchard |
Publisher | : Springer Science & Business Media |
Total Pages | : 469 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461200490 |
Physics has long been regarded as a wellspring of mathematical problems. Mathematical Methods in Physics is a self-contained presentation, driven by historic motivations, excellent examples, detailed proofs, and a focus on those parts of mathematics that are needed in more ambitious courses on quantum mechanics and classical and quantum field theory. Aimed primarily at a broad community of graduate students in mathematics, mathematical physics, physics and engineering, as well as researchers in these disciplines.
Author | : Carlo Alabiso |
Publisher | : Springer Nature |
Total Pages | : 343 |
Release | : 2021-03-03 |
Genre | : Science |
ISBN | : 3030674177 |
This book offers an essential introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for providing an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, lies in the strenuous mathematics demands that even the simplest physical cases entail. Graduate courses in physics rarely offer enough time to cover the theory of Hilbert space and operators, as well as distribution theory, with sufficient mathematical rigor. Accordingly, compromises must be found between full rigor and the practical use of the instruments. Based on one of the authors’s lectures on functional analysis for graduate students in physics, the book will equip readers to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. It also includes a brief introduction to topological groups, and to other mathematical structures akin to Hilbert space. Exercises and solved problems accompany the main text, offering readers opportunities to deepen their understanding. The topics and their presentation have been chosen with the goal of quickly, yet rigorously and effectively, preparing readers for the intricacies of Hilbert space. Consequently, some topics, e.g., the Lebesgue integral, are treated in a somewhat unorthodox manner. The book is ideally suited for use in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.
Author | : Edoardo Provenzi |
Publisher | : John Wiley & Sons |
Total Pages | : 368 |
Release | : 2021-08-03 |
Genre | : Mathematics |
ISBN | : 1119851300 |
From Euclidian to Hilbert Spaces analyzes the transition from finite dimensional Euclidian spaces to infinite-dimensional Hilbert spaces, a notion that can sometimes be difficult for non-specialists to grasp. The focus is on the parallels and differences between the properties of the finite and infinite dimensions, noting the fundamental importance of coherence between the algebraic and topological structure, which makes Hilbert spaces the infinite-dimensional objects most closely related to Euclidian spaces. The common thread of this book is the Fourier transform, which is examined starting from the discrete Fourier transform (DFT), along with its applications in signal and image processing, passing through the Fourier series and finishing with the use of the Fourier transform to solve differential equations. The geometric structure of Hilbert spaces and the most significant properties of bounded linear operators in these spaces are also covered extensively. The theorems are presented with detailed proofs as well as meticulously explained exercises and solutions, with the aim of illustrating the variety of applications of the theoretical results.
Author | : Edited by Paul F. Kisak |
Publisher | : Createspace Independent Publishing Platform |
Total Pages | : 190 |
Release | : 2016-01-07 |
Genre | : |
ISBN | : 9781523323999 |
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)-and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. This book gives a mathematical overview of the definition and use of Hilbert Space.
Author | : Sterling K. Berberian |
Publisher | : American Mathematical Soc. |
Total Pages | : 226 |
Release | : 1999 |
Genre | : Mathematics |
ISBN | : 0821819127 |
From the Preface: ``This textbook has evolved from a set of lecture notes ... In both the course and the book, I have in mind first- or second-year graduate students in Mathematics and related fields such as Physics ... It is necessary for the reader to have a foundation in advanced calculus which includes familiarity with: least upper bound (LUB) and greatest lower bound (GLB), the concept of function, $\epsilon$'s and their companion $\delta$'s, and basic properties of sequences of real and complex numbers (convergence, Cauchy's criterion, the Weierstrass-Bolzano theorem). It is not presupposed that the reader is acquainted with vector spaces ... , matrices ... , or determinants ... There are over four hundred exercises, most of them easy ... It is my hope that this book, aside from being an exposition of certain basic material on Hilbert space, may also serve as an introduction to other areas of functional analysis.''
Author | : Hwa-Long Gau |
Publisher | : Cambridge University Press |
Total Pages | : 556 |
Release | : 2021-08-05 |
Genre | : Mathematics |
ISBN | : 1108787606 |
Starting with elementary operator theory and matrix analysis, this book introduces the basic properties of the numerical range and gradually builds up the whole numerical range theory. Over 400 assorted problems, ranging from routine exercises to published research results, give you the chance to put the theory into practice and test your understanding. Interspersed throughout the text are numerous comments and references, allowing you to discover related developments and to pursue areas of interest in the literature. Also included is an appendix on basic convexity properties on the Euclidean space. Targeted at graduate students as well as researchers interested in functional analysis, this book provides a comprehensive coverage of classic and recent works on the numerical range theory. It serves as an accessible entry point into this lively and exciting research area.
Author | : N. Young |
Publisher | : Cambridge University Press |
Total Pages | : 254 |
Release | : 1988-07-21 |
Genre | : Mathematics |
ISBN | : 1107717167 |
This textbook is an introduction to the theory of Hilbert space and its applications. The notion of Hilbert space is central in functional analysis and is used in numerous branches of pure and applied mathematics. Dr Young has stressed applications of the theory, particularly to the solution of partial differential equations in mathematical physics and to the approximation of functions in complex analysis. Some basic familiarity with real analysis, linear algebra and metric spaces is assumed, but otherwise the book is self-contained. It is based on courses given at the University of Glasgow and contains numerous examples and exercises (many with solutions). Thus it will make an excellent first course in Hilbert space theory at either undergraduate or graduate level and will also be of interest to electrical engineers and physicists, particularly those involved in control theory and filter design.
Author | : Eduard Prugovecki |
Publisher | : Courier Corporation |
Total Pages | : 722 |
Release | : 2013-07-02 |
Genre | : Science |
ISBN | : 0486318052 |
A critical presentation of the basic mathematics of nonrelativistic quantum mechanics, this text is suitable for courses in functional analysis at the advanced undergraduate and graduate levels. Its readable and self-contained form is accessible even to students without an extensive mathematical background. Applications of basic theorems to quantum mechanics make it of particular interest to mathematicians working in functional analysis and related areas. This text features the rigorous proofs of all the main functional-analytic statements encountered in books on quantum mechanics. It fills the gap between strictly physics- and mathematics-oriented texts on Hilbert space theory as applied to nonrelativistic quantum mechanics. Organized in the form of definitions, theorems, and proofs of theorems, it allows readers to immediately grasp the basic concepts and results. Exercises appear throughout the text, with hints and solutions at the end.
Author | : Jonathan H. Manton |
Publisher | : |
Total Pages | : 138 |
Release | : 2015-11-20 |
Genre | : Technology & Engineering |
ISBN | : 9781680830927 |
Hilbert space theory is an invaluable mathematical tool in numerous signal processing and systems theory applications. Hilbert spaces satisfying certain additional properties are known as Reproducing Kernel Hilbert Spaces (RKHSs). This primer gives a gentle and novel introduction to RKHS theory. It also presents several classical applications. It concludes by focusing on recent developments in the machine learning literature concerning embeddings of random variables. Parenthetical remarks are used to provide greater technical detail, which some readers may welcome, but they may be ignored without compromising the cohesion of the primer. Proofs are there for those wishing to gain experience at working with RKHSs; simple proofs are preferred to short, clever, but otherwise uninformative proofs. Italicised comments appearing in proofs provide intuition or orientation or both. A Primer on Reproducing Kernel Hilbert Spaces empowers readers to recognize when and how RKHS theory can profit them in their own work.