Lectures:   MWF 2:30-3:20 pm online via Zoom

Instructor: Robert Pego   (Office: 6127 Wean Hall.   Phone: (412)268-2553.   Email: rpego AT cmu DOT edu )

• Office hours: online 4:00-6:00pm MWTh or let's arrange a time to meet

Assignments: (Future assignments will be posted on Canvas)

• Assignment 1, due F 1/31 pdf
• Assignment 2, due W 2/12 pdf
• Assignment 3, due W 2/26 pdf
• Assignment 4, now due W 3/25 pdf

Links:

• Proof of Tychonoff's theorem (based on Chernoff's 1992 proof, with an improvement by Pugh) link

Highly recommended reference works:

• N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Wiley Interscience. (Huge. All but essential.)
• T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag. (Especially good on spectral theory, applications to differential operators.)
• H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. (Expanded from the beautiful 1983 Analyse fonctionnelle, in easy French.)

• P. Lax, Functional Analysis, Wiley Interscience, 2002. (Interesting applications, viewpoint of a master mathematician.)
• W. Rudin, Functional Analysis, McGraw-Hill, 1973. (Elegant. Terse.)
• A. W. Naylor and G. R. Sell, Linear Operator Theory in Engineering and Science, Springer-Verlag, 1982. (More basic, tilted slightly toward science.)
• A. Friedman, Foundations of Modern Analysis, Dover, 1982. (Just the bare bones, but cheap.)

Course Overview:   Functional analysis is a large subject and this is only an introduction to a central set of topics. Research has largely tapered off in the subject itself, but it is widely used in many areas of application.

• Linear spaces: Hilbert spaces, Banach spaces, Topological vector spaces (esp. Schwartz' distributions).
• Hilbert spaces: geometry, projections, Riesz representation theorem, bilinear and quadratic forms, orthonormal sets and Fourier series.
• Banach spaces: continuity of linear mappings, Hahn-Banach, uniform boundedness, open-mapping theorems. Compact operators, unbounded operators, closed operators.
• Dual spaces: weak and weak-star topologies, reflexivity, convexity. Adjoints of operators: basic properties, null spaces and ranges. Sequences of bounded linear operators: weak, strong and uniform convergence.
• Solvability criteria for linear equations: spectrum and resolvent of bounded operators, spectral theory of compact operators, Fredholm alternative.
• Applications: important function spaces in analysis, differential operators, calculus of variations, evolution equations.

Learning objectives:   Students will gain understanding of fundamental concepts concerning complete linear spaces with norms and inner products, and maps between such, as indicated above. They will become familiar with the different topologies in common use for infinite-dimensional spaces and spaces of linear operators, and their relation to duality principles. They will master proofs of the fundamental theorems that characterize spaces and their topologies, maps and their spectrum. Also they will gain the ability to use the concepts, methods, and results of functional analysis to treat problems related to Fourier series, differential equations, evolution equations and other applications.

Prerequisites: Real analysis (21355), linear algebra (21341), metric-space topology (21465). (Course numbers are only examples.) Also useful, but not essential, would be some knowledge of complex variables and/or Fourier series.

Grading: Based on approx. 8 homework sets (35%), a midterm (25%) and a final exam (40%). Late homework will not receive a grade. The lowest homework score will be dropped, however.

Homework: PDF files will be posted online at http://www.math.cmu.edu/~bobpego/21640/ .

Academic integrity requires that your tests and homework solutions are your independent work and not copied from other sources. On homework you are encouraged, however, to discuss with others and consult other resources to improve your understanding.

Make-up tests are only possible in the case of a documented medical excuse, a university-sanctioned absence (e.g., participation in a varsity sporting event), or a family emergency. Please contact me at the earliest time possible to schedule a make-up.

Health and wellness: Graduate study is time-consuming and can be stressful. Make your health a priority, be smart about time management, seek social support, and ask for help when needed.