Functional Analysis, Spectral Theory, and Applications

This textbook provides a careful treatment of functional analysis and some of its applications in analysis, number theory, and ergodic theory.
In addition to discussing core material in functional analysis, the authors cover more recent and advanced topics, including Weyl's law for eigenfunctions of the Laplace operator, amenability and property (T), the measurable functional calculus, spectral theory for unbounded operators, and an account of Tao's approach to the prime number theorem using Banach algebras. The book further contains numerous examples and exercises, making it suitable for both lecture courses and self-study.

Functional Analysis, Spectral Theory, and Applications is aimed at postgraduate and advanced undergraduate students with some background in analysis and algebra, but will also appeal to everyone with an interest in seeing how functional analysis can be applied to other parts of mathematics.

Inhaltsverzeichnis

Motivation.- Norms and Banach Spaces.- Hilbert Spaces, Fourier Series, Unitary Representations.- Uniform Boundedness and Open Mapping Theorem.- Sobolev Spaces and Dirichlet's Boundary Problem.- Compact Self-Adjoint Operators, Laplace Eigenfunctions.- Dual Spaces.- Locally Convex Vector Spaces.- Unitary Operators and Flows, Fourier Transform.- Locally Compact Groups, Amenability, Property (T).- Banach Algebras and the Spectrum.- Spectral Theory and Functional Calculus.- Self-Adjoint and Symmetric Operators.- The Prime Number Theorem.- Appendix A: Set Theory and Topology.- Appendix B: Measure Theory.- Hints for Selected Problems.- Notes. 

Über den Autor / die Autorin

Menny Aka studied at the Hebrew University, with a Ph.D. in 2012 under Alexander Lubotzky. He held research positions at EPFL and ETH Zürich before becoming a senior scientist at ETH Zürich. He works on the interaction between number theory, ergodic theory and group theory. An enthusiastic and innovative lecturer, he is interested in making mathematics accessible, especially to younger audiences. He has initiated and taught in various programs for high school students, including projects aimed at gifted students and prospective undergraduates. He is interested in showcasing the beauty and simplicity underpinning complex mathematical ideas.

Manfred Einsiedler studied at the University of Vienna, with a Ph.D. in 1999 under Klaus Schmidt. He held research positions at the University of East Anglia, Penn State University, the University of Washington, and Princeton University as a Clay Research Scholar. After becoming a Professor at Ohio State University he joinedETH Zürich. In 2004 he won the Research Prize of the Austrian Mathematical Society, in 2008 he was an invited speaker at the European Mathematical Congress in Amsterdam, and in 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad. He works on ergodic theory (especially dynamical and equidistribution problems on homogeneous spaces) and its applications to number theory. He has collaborated with Grigory Margulis and Akshay Venkatesh. With Elon Lindenstrauss and Anatole Katok, Einsiedler proved that a conjecture of Littlewood on Diophantine approximation is "almost always" true.

Thomas Ward studied at the University of Warwick, with a Ph.D. in 1989 under Klaus Schmidt. He held research positions at the University of Maryland, College Park and at Ohio State University before joining the University of East Anglia in 1992. Since 2008 he has served on university executives, as Pro-Vice-Chancellor for Education at the University of East Anglia and Durham University, and since 2016 as Deputy Vice-Chancellor (Student Education) at the University of Leeds. He worked on the ergodic theory of algebraic dynamical systems, compact group automorphisms, and number theory. A long collaboration with Graham Everest on links between number theory and dynamical systems included the book "Heights of polynomials and entropy in algebraic dynamics" and a paper on Diophantine equations that won the 2012 Lester Ford Prize for mathematical exposition. With Einsiedler he has written "Ergodic theory with a view towards number theory" in 2011 and "Functional analysis, spectral theory, and applications" in 2017.

Zusammenfassung

This textbook provides a careful treatment of functional analysis and some of its applications in analysis, number theory, and ergodic theory.
In addition to discussing core material in functional analysis, the authors cover more recent and advanced topics, including Weyl’s law for eigenfunctions of the Laplace operator, amenability and property (T), the measurable functional calculus, spectral theory for unbounded operators, and an account of Tao’s approach to the prime number theorem using Banach algebras. The book further contains numerous examples and exercises, making it suitable for both lecture courses and self-study.

Functional Analysis, Spectral Theory, and Applications is aimed at postgraduate and advanced undergraduate students with some background in analysis and algebra, but will also appeal to everyone with an interest in seeing how functional analysis can be applied to other parts of mathematics.

Zusatztext

“All chapters end with a very useful list of additional topics and suggestions for further reading. The book also contains an appendix on set theory and topology, another one on measure theory … . The book is carefully written and provides an interesting introduction to functional analysis with a wealth of both classical and more recent applications.” (Michael M. Neumann, Mathematical Reviews, July, 2018)

“This is an attractive new textbook in functional analysis, aimed at … graduate students. … the large amount of material covered in this book … as well its overall readability, makes it useful as a reference as well as a potential graduate textbook. If you like functional analysis, teach it, or use it in your work, this book certainly merits a careful look.” (Mark Hunacek, MAA Reviews, January, 2018).

“The present book is different from the usual textbooks on functional analysis: it does not only cover the basic material, but also a number of advanced topics which cannot be found in many other books on the subject. … The text is suitable for self-study as well as for the preparation of lectures and seminars. … this is a highly recommendable book for students and researchers alike who are interested in functional analysis and its broad applications.” (Jan-David Hardtke, zbMATH 1387.46001, 2018)

Bericht

"All chapters end with a very useful list of additional topics and suggestions for further reading. The book also contains an appendix on set theory and topology, another one on measure theory ... . The book is carefully written and provides an interesting introduction to functional analysis with a wealth of both classical and more recent applications." (Michael M. Neumann, Mathematical Reviews, July, 2018)

"This is an attractive new textbook in functional analysis, aimed at ... graduate students. ... the large amount of material covered in this book ... as well its overall readability, makes it useful as a reference as well as a potential graduate textbook. If you like functional analysis, teach it, or use it in your work, this book certainly merits a careful look." (Mark Hunacek, MAA Reviews, January, 2018).

"The present book is different from the usual textbooks on functional analysis: it does not only cover the basic material, but also a number of advanced topics which cannot be found in many other books on the subject. ... The text is suitable for self-study as well as for the preparation of lectures and seminars. ... this is a highly recommendable book for students and researchers alike who are interested in functional analysis and its broad applications." (Jan-David Hardtke, zbMATH 1387.46001, 2018)

Produktdetails

• Autoren: Manfred Einsiedler, Thomas Ward
• Verlag: Springer, Berlin
• Sprache: Englisch
• Produktform: Taschenbuch
• Erschienen: 30.08.2018
• EAN: 9783319864235
• ISBN: 978-3-31-986423-5
• Seiten: 614
• Abmessung: 155 mm x 236 mm x 35 mm
• Gewicht: 949 g
• Illustration: XIV, 614 p. 33 illus.
• Serie: Graduate Texts in Mathematics
• Themen: Naturwissenschaften, Medizin, Informatik, Technik > Mathematik > Analysis; Analysis, Zahlentheorie, B, Kybernetik und Systemtheorie, Differentialrechnung und -gleichungen, Komplexe Analysis, komplexe Variablen, Funktionentheorie, Dynamics, Mathematics and Statistics, Functional Analysis, Number Theory, Ordinary Differential Equations, Dynamical Systems and Ergodic Theory, Partial Differential Equations, Complex analysis, complex variables, Ergodic theory, Abstract Harmonic Analysis, Nonlinear science, Harmonic analysis, Differential calculus & equations, Differential equations, Dynamical systems