April 2002, ©2002
* 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.
"For years Lax has been counted among the world's very top people in PDEs, so no serious student can afford to ignore his view of the foundations leading up to that subject." (Choice, Vol. 40, No. 4, December 2002)
"...attractive...well suited for graduate courses...and useful for research mathematicians." (Mathematical Reviews, 2003a)
"...The book is highly recommended to all students of analysis". (Zentralblatt MATH, Vol.1009, No.9, 2003)
"A lot of good material, doled out in short chapters." (American Mathematical Monthly, August/September 2003)