Mathematical Physics

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University of Chicago Press, 15.09.1985 - 351 Seiten
Mathematical Physics is an introduction to such basic mathematical structures as groups, vector spaces, topological spaces, measure spaces, and Hilbert space. Geroch uses category theory to emphasize both the interrelationships among different structures and the unity of mathematics. Perhaps the most valuable feature of the book is the illuminating intuitive discussion of the "whys" of proofs and of axioms and definitions. This book, based on Geroch's University of Chicago course, will be especially helpful to those working in theoretical physics, including such areas as relativity, particle physics, and astrophysics.

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Inhalt

Introduction
2
Categories
4
The Category of Groups
17
Subgroups
25
Normal Subgroups
30
Homomorphisms
33
Direct Products and Sums of Groups
36
Relations
40
Compactness
166
The CompactOpen Topology
173
Connectedness
178
Example Dynamical Systems
184
Homotopy
189
Homology
200
Homology Relation to Homotopy
212
The Homology Functors
215

The Category of Vector Spaces
45
Subspaces
54
Linear Mappings Direct Products and Sums
59
From Real to Complex Vector Spaces and Back
63
Duals
66
Multilinear Mappings Tensor Products
72
Example Minkowski Vector Space
80
Example The Lorentz Group
88
Functors
91
The Category of Associative Algebras
98
The Category of Lie Algebras
105
Example The Algebra of Observables
112
Example Fock Vector Space
115
Representations General Theory
121
Representations on Vector Spaces
126
The Algebraic Categories Summary
133
Subsets and Mappings
135
Topological Spaces
137
Continuous Mappings
148
The Category of Topological Spaces
155
Nets
161
Uniform Spaces
218
The Completion of a Uniform Space
226
Topological Groups
235
Topological Vector Spaces
241
Categories Summary
249
Measure Spaces
250
Constructing Measure Spaces
258
Measurable Functions
260
Integrals
263
Distributions
271
Hilbert Spaces
278
Bounded Operators
286
The Spectrum of a Bounded Operator
294
The Spectral Theorem Finitedimensional Case
303
Continuous Functions of a Hermitian Operator
307
Other Functions of a Hermitian Operator
312
The Spectral Theorem
320
Operators Not Necessarily Bounded
325
SelfAdjoint Operators
330
Index of Defined Terms
344
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Über den Autor (1985)

Robert Geroch is professor in the Departments of Physics and Mathematics and at the Enrico Fermi Institute at the University of Chicago.

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