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E-grāmata: Introduction to Functional Analysis

  • Formāts: EPUB+DRM
  • Sērija : Compact Textbooks in Mathematics
  • Izdošanas datums: 30-Nov-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030527846
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  • Formāts: EPUB+DRM
  • Sērija : Compact Textbooks in Mathematics
  • Izdošanas datums: 30-Nov-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030527846
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Functional analysis has become one of the essential foundations of modern applied mathematics in the last decades, from the theory and numerical solution of differential equations, from optimization and probability theory to medical imaging and mathematical image processing.This textbook offers a compact introduction to the theory and is designed to be used during one semester, fitting exactly 26 lectures of 90 minutes each. It ranges from the topological fundamentals recalled from basic lectures on real analysis to spectral theory in Hilbert spaces. Special attention is given to the central results on dual spaces and weak convergence.

Topological basics.- Linear operators between normed spaces.- Dual spaces and weak convergence.- Compact operators between Banach spaces.- Hilbert spaces.

Christian Clason is Professor at the Faculty of Mathematics of the University of Duisburg-Essen.

Recenzijas

The first remarkable aspect of this unique book is its unifying approach. The book also is well written, is self-contained, and includes timely, useful lists of references. Any researcher or graduate student working on fractional-in-time semilinear equations should read this essential book. (Pablo Raśl Stinga, SIAM Review, Vol. 64 (1), March, 2022)



In my next functional analysis course I will definitely include Clasons book in my reading list . this book is a valuable addition to the literature for anyone teaching a one-semester course in functional analysis. It covers all the important topics that we want graduate students to understand, and in a very structured and efficient way. Students may find it helpful to have an efficient and concise textbook for self-study and reference purposes. (Armin Schikorra, SIAM Review, Vol. 63 (4), December, 2021)



I do like the book! Most impressive to me is perhaps that the whole text is just 160 pages and still the explanations do not feel condensed while really the essential points are taken care of. (Olav Nygaard, zbMATH 1461.46001, 2021)

Part I Topological Basics
1 Metric Spaces
3(6)
2 Compact Sets
9(10)
Part II Linear Operators Between Normed Spaces
3 Normed Vector Spaces
19(16)
4 Linear Operators
35(10)
5 The Uniform Boundedness Principle
45(10)
6 Quotient Spaces
55(8)
Part III Dual Spaces and Weak Convergence
7 Linear Functionals and Dual Spaces
63(8)
8 The Hahn-Banach Theorem
71(12)
9 Adjoint Operators
83(10)
10 Reflexivity
93(6)
11 Weak Convergence
99(12)
Part IV Compact Operators Between Banach Spaces
12 Compact Operators
111(8)
13 The Fredholm Alternative
119(4)
14 The Spectrum
123(12)
Part V Hilbert Spaces
15 Inner Products and Orthogonality
135(14)
16 The Riesz Representation Theorem
149(8)
17 Spectral Decomposition in Hilbert Spaces
157(8)
References 165(2)
Index 167
Christian Clason is Professor at the Faculty of Mathematics of the University of Duisburg-Essen.