Instructor: Florent Baudier
Office: Blocker 525J
Office hours: M 5:30-6:30 p.m. or by appointment
Lectures: MW 4:10 p.m.-5:25 p.m. BLOC 110
Course description: Banach spaces, theorems of Hahn-Banach and Banach-Steinhaus, the closed graph and open mapping theorems, Hilbert spaces, topological vector spaces and weak topologies. Textbook:
Real Analysis (Modern Techniques and Their Applications), Second Edition, Gerald B. Folland Exams Homework   Schedule

Date of Class	Material covered
Wednesday 01/18	General Topology: definitions, examples, relative topology, convergence of sequences, continuity of function, interior, closure, boundary, accumulation points, dense, nowhere dense Separation axioms: T1, T2 (Hausdorff)
Monday 01/23	Local base, base, subbase, topology generated by a collection, local continuity
Wenesday 01/25	Neighborhoods, countability axioms: first countable, second countable, nets
Monday 01/30	cluster points, limit points, Kelley subnets, cofinal subnets
Wednesday 02/01	class canceled due to inclement weather
Monday 02/06	compactness
Wednesday 02/08	normal spaces, Urysohn Lemma, Tietze extension theorem, initial topologies
Monday 02/13	product topology, Tychonoff compactness theorem, Arzela-Ascoli
Wednesday 02/15	Class does not meet
Monday 02/20	Stone-Weierstrass theorem(s)
Wednesday 02/22	Baire spaces, equivalent characterizations, applications, Baire Category Theorem, meager sets, normed vector spaces, Banach spaces
Monday 02/27	continuous linear maps, completness criterion, completness of L(X,Y), weak and strong Uniform Boundedness Principle, Banach-Steinhaus theorem
Wednesday 03/01	Isomorphisms, open maps, Open Mapping Theorem, Bounded Inverse Theorem, sums of Banach spaces, Closed Graph Theorem
Monday 04/06 (extended lecture)	Linear functionals, dual space, algebraic/analytic Hahn-Banach Theorem(s) and consequences, locally convex topological vector spaces, topologies generated by semi-norms, definitions of weak and weak* topologies
Wednesday 03/08	MIDTERM,
Monday 03/13	Spring Break
Wednesday 03/15	Spring Break
Monday 03/20	Alaoglu's theorem, useful algebraic lemmas, important facts about the weak and weak* topologies, gauge of a convex, geometric forms of the Hahn-Banach theorem, applications
Wednesday 03/22	Lp-spaces, Holder's inequality, Minkowski's inequality, completness of Lp
Monday 03/27	density of simple functions, L&infty;-spaces, sums and intersections of Lp-spaces, interpolation inequality
Wednesday 03/29	Comparing Lp-spaces, Hilbert spaces, orthogonality, Pythagorean theorem, Cauchy-Schwarz inequality, parallelogram law, polarization identity
Monday 04/03	Bessel's inequality, Parseval's identity, orthogonal bases, unitary operators,
Wednesday 04/05	linear vs bounded linear projections, complemented subspaces, orthogonal projections, duality of Hilbert spaces
Monday 04/10	duality of Lp-spaces, weak Lp-spaces
Wednesday 04/12	Interpolation: Riesz-Thorin interpolation theorem, Marcinkiewicz interpolation theorem
Monday 04/17	Class does not meet
Wednesday 04/19	Class does not meet
Monday 04/24 (extended lecture)	bounded linear functional on the Banach space of complex-valued continuous functions with compact support or vanishing at infinity on a locally compact Hausdorff space: reduction to real-functionals on the subspace of real-valued functions, Jordan decomposition of linear functionals
Wednesday 04/26 (extended lecture)	positive Radon measures, subordinated partition of unity, Riesz-Markov-Kakutani representation of positive linear functionals.
Monday 05/01	Complex measures, polar decomposition, total variation norm, Banach space of complex Radon measures, Lusin's aproximation theorem, C0(X)*=Radon(X)
Wednesday 05/03	Reading Day, No Classes
Friday 05/05	FINAL EXAM 3:30-5:30 pm