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source: University of Nottingham 上次更新日期：2014年6月28日

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; -- basic properties of Banach algebras.

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

Additional materials for this module are available at: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=... and http://itunesu.nottingham.ac.uk/albums/64.rss. See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/

Lecture 1: 35:10

Lecture 2: Complete metric spaces 41:30

Lecture 3: revision of Metric and Topological Spaces 44:33

Lecture 4: Complete metric spaces continued 45:13

Lecture 5a: Nowhere dense sets 32:31

Lecture 5b: Infinite products and Tychonoff's theorem 10:44

Lecture 6a: Discussion session on partially ordered sets and vector spaces 12:23

Lecture 6b: Continuation of discussion session on partially ordered sets 32:40

Lecture 7: Infinite products and Tychonoff's theorem 48:06

Lecture 8: The proof of Tychonoff's theorem 44:56

Lecture 9a: Infinite products and Tychonoff's theorem 26:15

Lecture 9b: Normed spaces and Banach spaces 12:54

Lecture 10: Normed spaces and Banach spaces 47:12

Lecture 11a: 26:41

Lecture 11b: 12:18

Lecture 12: Normed spaces and Banach spaces 43:20

Lecture 13a: Normed spaces and Banach spaces 9:02

Lecture 13b: Equivalence of norms 31:45

Lecture 14a: A recap of Equivalence of norms 3:49

Lecture 14b: A recap of Equivalence of norms 47:27

Lecture 15b: Linear Maps 35:05

Lecture 16a: Linear maps 24:43

Lecture 16b: Sequence spaces 17:45

Lecture 15a: Final discussion of Equivalence of norms 10:08

Lecture 17: Sequence spaces continued 46:31

Lecture 18a: more about Sequence spaces 36:01

Lecture 18b: Isomorphisms 7:08

Lecture 19a: Isomorphisms of normed spaces 24:50

Lecture 19b: Sums and Quotients of Vector Spaces 17:12

Lecture 20a - Sums and quotients of vector spaces - Conclusion of Section 3.6, printed slides 57-60 22:07

Lecture 20b - Section 3.7: Dual spaces - Section 3.7, printed slides 61-64 22:41

Lecture 21 - Duals and Double Duals - Section 3.7, printed slides 64-65 35:38

Lecture 22 - Conclusion of Section 3.7 and brief introduction to Section 3.8 41:20

Lecture 23 - Extensions of linear maps 45:54

Lecture 24 - Completions, quotients and Riesz's Lemma 40:45

Lecture 25 - The Weak-* Topology and the Banach-Alaoglu Theorem 42:58

Lecture 26 - Open Mappings and their Applications 38:30

Lecture 27 - Applications of the open mapping lemma 40:01

Lecture 28 - part a - Recap concerning convex sets which are symmetric about 0 2:04

Lecture 28 - part b - Chapter 5, printed slides 102-108, Proof of Open Mapping Theorem 44:29

Lecture 29 part a - Recap, and proof of the Closed Graph Theorem 18:15

Lecture 29 part b - The Uniform Boundedness Principle/Banach-Steinhaus 21:40

Lecture 30 - Commutative Banach Algebras, printed slides 1-19 47:33

Lecture 31 - Commutative Banach Algebras, printed slides 19-29 46:15

Lecture 32 - Discussion session on measure theory 46:16

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