Introduction to Functional Analysis and Its Applications
Buch, Englisch, 368 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 721 g
ISBN: 978-1-78630-682-1
Verlag: Wiley
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.
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Preface xi
Chapter 1. Inner Product Spaces (Pre-Hilbert) 1
Chapter 2. The Discrete Fourier Transform and its Applications to Signal and Image Processing 31
Chapter 3. Lebesgue's Measure and Integration Theory 105
Chapter 4. Banach Spaces and Hilbert Spaces 115
Chapter 5. The Geometric Structure of Hilbert Spaces 171
Chapter 6. Bounded Linear Operators in Hilbert Spaces 221
Appendix 1: Quotient Space 323
Appendix 2: The Transpose (or Dual) of a Linear Operator 329
Appendix 3: Uniform, Strong and Weak Convergence 331
References 335
Index 337
