infocobuild

G14FUN - Functional Analysis

G14FUN - Functional Analysis (University of Nottingham). This is a collection of video lectures taught by Dr. Joel Feinstein. Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions. This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include: norm topology and topological isomorphism; boundedness of operators; compactness and finite dimensionality; extension of functionals; weak*-compactness; sequence spaces and duality; and basic properties of Banach algebras. (from unow.nottingham.ac.uk)

Functional Analysis


Lecture 01 - Functional Analysis - Totally Ordered Sets and Partially Ordered Sets
Lecture 02 - Complete Metric Spaces
Lecture 03 - Revision of Metric and Topological Spaces
Lecture 04 - Complete Metric Spaces II - Proof of the Baire Category Theorem
Lecture 5a - Nowhere Dense Sets
Lecture 5b - Infinite Products and Tychonoff's Theorem
Lecture 6a - Discussion Session on Partially Ordered Sets and Vector Spaces
Lecture 6b - Discussion Session on Partially Ordered Sets and Vector Spaces (cont.)
Lecture 07 - Infinite Products and Tychonoff's Theorem
Lecture 08 - The Proof of Tychonoff's Theorem
Lecture 9a - Infinite Products and Tychonoff's Theorem
Lecture 9b - Normed Spaces and Banach Spaces
Lecture 10 - Normed Spaces and Banach Spaces
Lecture 11a - Completeness of the Uniform Norm
Lecture 11b - A Revision Interlude on Pointwise and Uniform Convergence for Sequences of Functions
Lecture 12 - Normed Spaces and Banach Spaces
Lecture 13a - Normed Spaces and Banach Spaces
Lecture 13b - Equivalence of Norms
Lecture 14a - A Recap of Equivalence of Norms
Lecture 14b - A Recap of Equivalence of Norms
Lecture 15a - Final Discussion of Equivalence of Norms
Lecture 15b - Linear Maps
Lecture 16a - Linear Maps and Connections with Lipschitz Continuity
Lecture 16b - Sequence Spaces
Lecture 17 - Sequence Spaces (cont.)
Lecture 18a - More about Sequence Spaces
Lecture 18b - Isomorphisms
Lecture 19a - Isomorphisms of Normed Spaces
Lecture 19b - Sums and Quotients of Vector Spaces
Lecture 20a - Sums and Quotients of Vector Spaces (cont.)
Lecture 20b - Dual Spaces
Lecture 21 - Duals and Double Duals
Lecture 22 - Conclusion of Dual Spaces
Lecture 23 - Extensions of Linear Maps
Lecture 24 - Completions, Quotients, and Riesz's Lemma
Lecture 25 - The Weak-* Topology and the Banach-Alaoglu Theorem
Lecture 26 - Open Mappings and Their Applications
Lecture 27 - Applications of the Open Mappings Lemma
Lecture 28a - Recap Concerning Convex Sets which are Symmetric about 0
Lecture 28b - Recap, Proof of the Opening Mapping Theorem
Lecture 29a - Recap, and Proof of the Closed Graph Theorem
Lecture 29b - The Uniform Boundedness Principle/Banach-Steinhaus
Lecture 30 - Commutative Banach Algebras
Lecture 31 - Commutative Banach Algebras
Lecture 32 - Discussion Session on the Measure Theory

References
Functional Analysis
View Resource. Download Resource. The complete set of videos and lecture notes can be viewed online or downloaded as a zip file. As taught Autumn semester 2010. Dr Joel Feinstein, School of Mathematical Sciences.