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Includes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more.
* Assumes prior knowledge of Naive set theory, linear algebra, point set topology, basic complex variable, and real variables.
* Includes an appendix on the Riesz representation theorem.
The Hahn-Banach Theorem.
Applications of the Hahn-Banach Theorem.
Normed Linear Spaces.
Applications of Hilbert Space Results.
Duals of Normed Linear Space.
Applications of Duality.
Applications of Weak Convergence.
The Weak and Weak* Topologies.
Locally Convex Topologies and the Krein-Milman Theorem.
Examples of Convex Sets and their Extreme Points.
Bounded Linear Maps.
Examples of Bounded Linear Maps.
Banach Algebras and their Elementary Spectral Theory.
Gelfand's Theory of Commutative Banach Algebras.
Applications of Gelfand's Theory of Commutative Banach Algebras.
Examples of Operators and their Spectra.
Examples of Compact Operators.
Positive Compact Operators.
Fredholm's Theory of Integral Equations.
Harmonic Analysis on a Halfline.
Compact Symmetric Operators in Hilbert Space.
Examples of Compact Symmetric Operators.
Trace Class and Trace Formula.
Spectral Theory of Symmetric, Normal and Unitary Operators.
Spectral Theory of Self-Adjoint Operators.
Examples of Self-Adjoint Operators.
Semigroups of Operators.
Groups of Unitary Operators.
Examples of Strongly Continuous Semigroups.
A Theorem of Beurling.
Appendix A: The Riesz-Kakutani Representation Theorem.
Appendix B: Theory of Distributions.
Appendix C: Zorn's Lemma.
Posted August 1, 2002
The subject of functional analysis, while fundamental and central in the vast landscape of all of mathematics, really started with seminal theorems due to Banach, Hilbert, the Riesz brothers, Fejer, von Neumann, Herglotz, Hausdorff, Friedrichs, Steinhouse,...and many other of, the perhaps less well known, founding fathers,-- in Central Europe (at the time), in the period between the two World Wars. The subject then gained from there, because of its many sucess stories,-- in proving new theorems, in unifying old ones, in offering a framework for quantum theory, for dynamical systems, for Fourier analysis, for approximation theory, and for partial differential equations. The Journal of Functional Analysis, starting in the 1960ties, broadened the subject, reaching almost all branches of science, and finding functional analytic flavor in theories surprisingly far from the original roots of the subject. Peter Lax has himself,-- alone and with others, shaped some of greatest successes of the period (scattering theory is but one of them),-- right up to the present. That is in the book!! And it offers an upbeat outlook for the future. It has been tested in the class room,-it is really user-friendly. At the end of each chapter P Lax ofers personal recollections;-- little known stories of how several of the pioneers in the subject have been victims,- in the 30ties and the 40ties, of Nazi atrocities. The writing is crisp and engaged,- the exercises are great;- just right for students to learn from, be it by selfstudy, or in class. This is the book to teach from!Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.