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Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups 2015 ed. [Kõva köide]

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  • Formaat: Hardback, 255 pages, kõrgus x laius: 235x155 mm, kaal: 5384 g, 5 Illustrations, black and white; XVII, 255 p. 5 illus., 1 Hardback
  • Sari: UNITEXT for Physics
  • Ilmumisaeg: 20-Oct-2014
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319037129
  • ISBN-13: 9783319037127
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Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups 2015 ed.Suurem pilt
  • Formaat: Hardback, 255 pages, kõrgus x laius: 235x155 mm, kaal: 5384 g, 5 Illustrations, black and white; XVII, 255 p. 5 illus., 1 Hardback
  • Sari: UNITEXT for Physics
  • Ilmumisaeg: 20-Oct-2014
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319037129
  • ISBN-13: 9783319037127
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319037127.html
Märksõnad:

This book is an 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 an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, resides in the very high mathematical difficulty of even the simplest physical case. Within an ordinary graduate course in physics there is insufficient time to cover the theory of Hilbert spaces and operators, as well as distribution theory, with sufficient mathematical rigor. Compromises must be found between full rigor and practical use of the instruments. The book is based on the author's lessons on functional analysis for graduate students in physics. It will equip the reader to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude.

With respect to the original lectures, the mathematical flavor in all subjects has been enriched. Moreover, a brief introduction to topological groups has been added in addition to exercises and solved problems throughout the text. With these improvements, the book can be used in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.



This introduction to Hilbert space will equip the reader for more demanding further study. It builds a pedagogic bridge between the rigorous style of mathematics and the more practical perspective of physicists.

Arvustused

This book introduces essential ideas and results of Hilbert space and linear operator theory to beginners in quantum physics. The textbook is suitable for advanced undergraduate or introductory graduate courses for both physics and mathematics students. (Michael Frank, Mathematical Reviews, August, 2015)

1 Introduction and Preliminaries
1(22)
1.1 Hilbert Space Theory---A Quick Overview
1(6)
1.1.1 The Real Numbers---Where it All Begins
2(3)
1.1.2 Linear Spaces
5(1)
1.1.3 Topological Spaces
6(1)
1.1.4 Metric Spaces
6(1)
1.1.5 Normed Spaces and Banach Spaces
6(1)
1.1.6 Topological Groups
7(1)
1.2 Preliminaries
7(16)
1.2.1 Sets
7(1)
1.2.2 Common Sets
7(1)
1.2.3 Relations Between Sets
8(1)
1.2.4 Families of Sets; Union and Intersection
8(1)
1.2.5 Set Difference, Complementation, and De Morgan's Laws
9(1)
1.2.6 Finite Cartesian Products
9(1)
1.2.7 Functions
10(1)
1.2.8 Arbitrary Cartesian Products
11(1)
1.2.9 Direct and Inverse Images
11(1)
1.2.10 Indicator Functions
12(1)
1.2.11 Cardinality
12(1)
1.2.12 The Cantor-Shroder-Bernstein Theorem
13(1)
1.2.13 Countable Arithmetic
14(1)
1.2.14 Relations
15(1)
1.2.15 Equivalence Relations
15(1)
1.2.16 Ordered Sets
16(1)
1.2.17 Zorn's Lemma
17(1)
1.2.18 A Typical Application of Zorn's Lemma
17(2)
1.2.19 The Real Numbers
19(2)
References
21(2)
2 Linear Spaces
23(52)
2.1 Linear Spaces---Elementary Properties and Examples
24(7)
2.1.1 Elementary Properties of Linear Spaces
25(2)
2.1.2 Examples of Linear Spaces
27(4)
2.2 The Dimension of a Linear Space
31(10)
2.2.1 Linear Independence, Spanning Sets, and Bases
31(3)
2.2.2 Existence of Bases
34(2)
2.2.3 Existence of Dimension
36(5)
2.3 Linear Operators
41(9)
2.3.1 Examples of Linear Operators
42(1)
2.3.2 Algebra of Operators
43(2)
2.3.3 Isomorphism
45(5)
2.4 Subspaces, Products, and Quotients
50(12)
2.4.1 Subspaces
50(4)
2.4.2 Kernels and Images
54(2)
2.4.3 Products and Quotients
56(3)
2.4.4 Complementary Subspaces
59(3)
2.5 Inner Product Spaces and Normed Spaces
62(13)
2.5.1 Inner Product Spaces
63(2)
2.5.2 The Cauchy-Schwarz Inequality
65(2)
2.5.3 Normed Spaces
67(1)
2.5.4 The Family of P Spaces
68(3)
2.5.5 The Family of Pre-Lp Spaces
71(3)
References
74(1)
3 Topological Spaces
75(38)
3.1 Topology---Definition and Elementary Results
76(13)
3.1.1 Definition and Motivation
76(5)
3.1.2 More Examples
81(1)
3.1.3 Elementary Observations
82(2)
3.1.4 Closed Sets
84(2)
3.1.5 Bases and Subbases
86(3)
3.2 Subspaces, Point-Set Relationships, and Countability Axioms
89(7)
3.2.1 Subspaces and Point-Set Relationships
89(3)
3.2.2 Sequences and Convergence
92(2)
3.2.3 Second Countable and First Countable Spaces
94(2)
3.3 Constructing Topologies
96(6)
3.3.1 Generating Topologies
97(2)
3.3.2 Coproducts, Products, and Quotients
99(3)
3.4 Separation and Connectedness
102(5)
3.4.1 The Hausdorff Separation Property
102(1)
3.4.2 Path-Connected and Connected Spaces
103(4)
3.5 Compactness
107(6)
References
111(2)
4 Metric Spaces
113(36)
4.1 Metric Spaces---Definition and Examples
114(5)
4.2 Topology and Convergence in a Metric Space
119(7)
4.2.1 The Induced Topology
120(3)
4.2.2 Convergence in Metric Spaces
123(3)
4.3 Non-Expanding Functions and Uniform Continuity
126(4)
4.4 Complete Metric Spaces
130(14)
4.4.1 Complete Metric Spaces
130(6)
4.4.2 Banach's Fixed-Point Theorem
136(1)
4.4.3 Baire's Theorem
137(2)
4.4.4 Completion of a Metric Space
139(5)
4.5 Compactness and Boundedness
144(5)
References
147(2)
5 Normed Spaces
149(50)
5.1 Semi-Norms, Norms, and Banach Spaces
150(16)
5.1.1 Semi-Norms and Norms
150(5)
5.1.2 Banach Spaces
155(3)
5.1.3 Bounded Operators
158(3)
5.1.4 The Open Mapping Theorem
161(2)
5.1.5 Banach Spaces of Linear and Bounded Operators
163(3)
5.2 Fixed-Point Techniques in Banach Spaces
166(8)
5.2.1 Systems of Linear Equations
166(2)
5.2.2 Cauchy's Problem and the Volterra Equation
168(2)
5.2.3 Fredholm Equations
170(4)
5.3 Inverse Operators
174(9)
5.3.1 Existence of Bounded Inverses
174(3)
5.3.2 Fixed-Point Techniques Revisited
177(6)
5.4 Dual Spaces
183(11)
5.4.1 Linear Functionals and the Riesz Representation Theorem
183(4)
5.4.2 Duals of Classical Spaces
187(3)
5.4.3 The Hahn-Banach Theorem
190(4)
5.5 Unbounded Operators and Locally Convex Spaces
194(5)
5.5.1 Closed Operators
194(2)
5.5.2 Locally Convex Spaces
196(2)
References
198(1)
6 Topological Groups
199(16)
6.1 Groups and Homomorphisms
200(3)
6.2 Topological Groups and Homomorphism
203(2)
6.3 Topological Subgroups
205(2)
6.4 Quotient Groups
207(2)
6.5 Uniformities
209(6)
References
213(2)
7 Solved Problems
215(36)
7.1 Linear Spaces
215(1)
7.2 Topological Spaces
216(2)
7.3 Metric Spaces
218(2)
7.4 Normed Spaces and Banach Spaces
220(2)
7.5 Topological Groups
222(1)
7.6 Solutions
223(28)
7.6.1 Linear Spaces
223(5)
7.6.2 Topological Spaces
228(7)
7.6.3 Metric Spaces
235(4)
7.6.4 Normed Spaces and Banach Spaces
239(5)
7.6.5 Topological Groups
244(7)
Index 251
Prof. Carlo Alabiso obtained his Degree in Physics at Milan University and then taught at Parma University, Parma, Italy for more than 40 years (with a period spent as a research fellow at the Stanford Linear Accelerator Center and at Cern, Geneva). His teaching encompassed topics in quantum mechanics, special relativity, field theory, elementary particle physics, mathematical physics, and functional analysis. His research fields include mathematical physics (Pad\'{e} approximants), elementary particle physics (symmetries and quark models), and statistical physics (ergodic problems), and he has published articles in a wide range of national and international journals as well as the previous Springer book (with Alessandro Chiesa), Problemi di Meccanica Quantistica non Relativistica.

Dr. Ittay Weiss completed his BSc and MSc studies in Mathematics at the Hebrew University of Jerusalem and he obtained his PhD in mathematics from Universiteit Utrecht in the Netherlands. He spent an additional three years in Utrecht as an assistant professor of mathematics, teaching mathematics courses across the entire undergraduate spectrum both at Utrecht University and at the affiliated University College Utrecht. He is currently a mathematics lecturer at the University of the South Pacific. His research interests lie in the fields of algebraic topology and operad theory, as well as the mathematical foundations of analysis and generalizations of metric spaces.