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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)

Lecture 5a - Nowhere Dense Sets

Go to the Course Home or watch other lectures:

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