## Math 581 - Functional Analysis I

Topics: Introduction to Hilbert and Banach spaces, linear operators, spectral theory, and, time permitting, applications to approximation theory and differential equations.

Prerequisites: Advanced calculus and linear algebra are required; Math 510 (introduction to analysis) is recommended.

Optional Textbook:
Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1st edition, 1989, ISBN-10: 0471504599.

 Meeting: MWF 2 - 2:50 p.m., in SMLC 352 Office hours: MWF 3:00 - 3:50 pm and also by appointment, in SMLC 212 Grades will be determined by homework
Homework:   HW1,   HW2,   HW3,   HW4,   HW5,   HW6,   HW7,   HW8,   HW9

Schedule
Jan 22, 24               Introduction. Overview of metric and normed spaces. (Chapters 1 and 2)
Jan 27, 29, 31         Sequence space ℓp: Minkowski inequality, Schauder basis, completeness. Compactness in metric spaces. Continuous and bounded mappings. (1.5, Chapter 2)
Feb 3, 5, 7              Spaces C[0,1] and Lp[0,1]. Hilbert spaces. (1.5, 3.1, 3.2)
Feb 10, 12, 14        Orthogonal complements, orthonormal sequences. Minimizing vectors in ℓ2. Series. (3.3, 3.4, 3.5)
Feb 17, 19, 21        Orthonormal basis. Basis in L2[0,1]. Isomorphism of Hilbert spaces. Representations of functionals on Hilbert spaces. Adjoint of an operator. (3.6, 3.7, 3.8, 3.9)
Feb 24, 26, 28        Properties of an adjoint operator. Self-adjoint and unitary operators. Completeness of the space of bounded linear operators. Spectrum. (3.9, 3.10, 2.10-2, 7.2)
Mar 3, 5, 7             Properties of resolvent and spectrum of a bounded operator. (7.3, 7.4, 7.5, 9.1)
Mar 10, 12, 14       Partially ordered sets. Hahn-Banach theorem. (4.1, 4.2, 4.3)
Mar 24, 26, 28       Dual spaces of ℓp, Lp[0,1], and C[0,1]. (2.10, 4.4)
Mar 31, Apr 2, 4    Adjoint operator. Reflexive spaces. (4.5, 4.6),  NMAS lecture.
Apr 7, 9, 11           Baire's category. Uniform boundedness principle. Strong and weak convergence (4.7, 4.8). Weak convergence in reflexive spaces.
Apr 14, 16, 18       Convergence of sequences of operators. Open mapping theorem. Closed operators. (4.9, 4.12, 4.13)
Apr 21, 23, 25       Closed graph theorem. Fixed point theorem and its applications. Spectral family. (4.13, 5.1, 5.2, 5.3, 5.4, 9.7)
Apr 28, 30, May 1  Properties of orthogonal projections. Positive operators. (9.5, 9.6, 9.3)
May 5, 7, 9           Square root of a positive operator. Spectral theorem. (9.4, 9.8, 9.9)