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This comprehensive introduction to functional analysis covers both the abstract theory and applications to spectral theory, the theory of partial differential equations, and quantum mechanics. It starts with the basic results of the subject and progresses towards a treatment of several advanced topics not commonly found in functional analysis textbooks, including Fredholm theory, form methods, boundary value problems, semigroup theory, trace formulas, and a mathematical treatment of states and observables in quantum mechanics. The book is accessible to graduate students with basic knowledge of topology, real and complex analysis, and measure theory. With carefully written out proofs, more than 300 problems, and appendices covering the prerequisites, this self-contained volume can be used as a text for various courses at the graduate level and as a reference text for researchers in the field.
Preface.
Notation and Conventions.
Banach Spaces.
The Classical Banach Spaces.
Hilbert Spaces.
Duality.
Bounded Operators.
Spectral Theory.
Compact Operators.
Bounded Operators on Hilbert Spaces.
The Spectral Theorem for Bounded Normal Operators.
The Spectral Theorem for Unbounded Normal Operators.
Boundary Value Problems.
Forms.
Semigroups of Linear Operators.
Trace Class Operators.
States and Observables.
Appendix A Zorn’s Lemma.
Appendix B Tensor Products.
Appendix C Topological Spaces.
Appendix D Metric Spaces.
Appendix E Measure Spaces.
Appendix F Integration.
Appendix G Notes.
References.
Index