Brief description: An introductory course in linear Functional Analysis.
After some Preliminaries (integration, various spaces, properties, examples) we will cover the basics on Banach spaces (continuous linear functionals and transformations; Hahn-Banach extension theorem; duality, weak convergence; Baire theorem, uniform boundedness; Open Mapping, Closed Graph, and Closed Range theorems; compactness; spectrum, Fredholm alternative), Hilbert spaces (orthogonality, bases, projections; Bessel and Parseval relations; Riesz representation theorem; spectral theory for compact, self-adjoint and normal operators; Sturm-Liouville theory), and Distributions (seminorms and locally convex spaces; test functions, distributions; calculus with distributions; etc.) with examples and applications. These are roughly the topics listed on the Applied Math. Course Syllabus

Textbook:   None

Some References:
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978.
J.B. Conway, A Course in Functional Analysis, Springer, 1990.
W. Rudin, Functional Analysis, McGraw-Hill, 1973.
M. Reed, B. Simon, Functional Analysis, Academic Press, 1980.
R. Meise, D. Vogt, Introduction to Functional Analysis, Oxford University Press, 1997.
K. Yosida, Functional Analysis, Springer, 1980.
H.L. Royden, Real Analysis, MacMillan, 1988.
S. Lang, Real Analysis, Addison Wesley, 1983.
T. Arbogast, J. Bona, Methods of Applied Mathematics, Course Notes, 2005.

Prerequisites: Knowledge of the subjects taught in the undergraduate analysis course M365C and an undergraduate course in linear algebra.

Consent of Instructor:   Not required for graduate students

First Day Handout:   Here

Homework:   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Some HW solutions:   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Exams:    old:  1, 2, 3    new:  1, 2, 3

Basics about metric spaces:   Notes here

Sketch of Lebesgue integration:   Notes here

Some inequalities:   Notes here

More References:
J. Hunter, B. Nachtergaele, Applied Analysis, World Scientific, 2001. Online, 2005.
S. Serfati, Functional Analysis Notes, Course Notes, 2004.
D.N. Arnold, Functional Analysis, Course Notes, 1997.
P. Garrett, Functional Analysis, Course Notes, 1996-2008.
W.W.L. Chen, Linear Functional Analysis, Course Notes, 1983-2001.
C. Remling, Functional Analysis, Lecture Notes, 2008-2009.
J. Schenker, Functional Analysis, Lecture Notes for Spring '08.
D. Hundertmark, Functional Analysis, Lecture Notes WS 2012/13.
M. Einsiedler, T. Ward, Functional Analysis Notes 2012.
A.C.R. Belton, Functional Analysis Course Notes 2004-2006.
A. Ulikowska, Lecture Notes in Functional Analysis.
G. Teschl, Topics in Real and Functional Analysis.
V. Gelfreich, Functional Analysis I, Lecture Notes 2010-2011.
R. Gelca, Functional Analysis Lecture Notes.
L. Gross, Lecture Notes on Functional Analysis, 2012.
T. Schlumprecht, Course Notes for Functional Analysis I, 2011.
R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, Online Book, 1994.
P. Cannarsa, T. D'Aprile, Lecture Notes on Measure Theory and Functional Analysis, 2006-2007.
L. Erdos, Notes on Integration and Fourier transform, 2004.
V. Liskevich, Measure Theory, Course Notes, 1998.
B.K. Driver, Measure Theory (Lecture Notes), 2000.
D.H. Sattinger, Measure Theory & Integration, 2004.
R.F. Bass, Real Analysis for Graduate Students: Measure and Integration, 2011.
W.P. Ziemer, Modern Real Analysis.
J.K. Hunter, Measure Theory, 2011.