Functional Analysis: Recent Episodes
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; – basic properties of Banach algebras.
Suitable for: Undergraduate students Level Four
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
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.
Chapter 5, printed slides 102-108, Proof of Open Mapping Theorem. (*note a 'typo' in the proof of Lemma 5.8 where one A should be an E)
Duals and Double Duals - Section 3.7, printed slides 64-65. Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Section 3.7, printed slides 64-65. Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Section 3.7: Dual spaces - Section 3.7, printed slides 61-64.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Section 3.7: Dual spaces - Section 3.7, printed slides 61-64.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sums and quotients of vector spaces - Conclusion of Section 3.6, printed slides 57-60.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sums and quotients of vector spaces - Conclusion of Section 3.6, printed slides 57-60.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sums and quotients of vector spaces - Section 3.6, printed slides 55-57.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sums and quotients of vector spaces - Section 3.6, printed slides 55-57.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Isomorphisms - Conclusion of Section 3.5: printed slides 51-54.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Isomorphisms - Conclusion of Section 3.5: printed slides 51-54.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Isomorphisms - Introduction to Section 3.5: printed slides 49-51.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Isomorphisms - Introduction to Section 3.5: printed slides 49-51.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sequence spaces - Conclusion of Section 3.4.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sequence spaces - Conclusion of Section 3.4,
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sequence spaces - Recap of Section 3.4 and proof of part of Theorem 3.15.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sequence spaces - Section 3.4, printed slides 45-48.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Sequence spaces - Section 3.4, printed slides 45-48.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Linear maps - Conclusion of Section 3.3 including recap, printed slide 44, connections with Lipschitz continuity
Linear maps - Conclusion of Section 3.3 including recap, printed slide 44, connections with Lipschitz continuity.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Linear maps - Section 3.3, printed slides 41-43.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Linear maps - Section 3.3, printed slides 41-43.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - final discussion of Section 3.2.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - final discussion of Section 3.2.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - "conclusion" of Section 3.2, including the proofs of Theorems 3.8 and 3.10.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - "conclusion" of Section 3.2, including the proofs of Theorems 3.8 and 3.10.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - recap of preliminary discussion relating to Theorem 3.8.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - recap of preliminary discussion relating to Theorem 3.8.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - whole section discussed, including discussion of facts needed to prove Theorem 3.8.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Equivalence of norms - whole section discussed, including discussion of facts needed to prove Theorem 3.8.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Normed spaces and Banach spaces - final discussion of Section 3.1.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Normed spaces and Banach spaces - final discussion of Section 3.1.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Normed spaces and Banach spaces - printed slides 33-34 concluded, and printed slides 35-37.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Normed spaces and Banach spaces - printed slides 33-34 concluded, and printed slides 35-37.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Lecture 10 - Normed spaces and Banach spaces - printed slides 31-34.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Infinite products and Tychonoff's theorem - printed slide 26 to end of chapter.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Normed spaces and Banach spaces - printed slides 28-31.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
The proof of Tychonoff's theorem.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Discussion session on partially ordered sets.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Infinite products and Tychonoff's theorem.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Chapter 2: Infinite products and Tychonoff’s theorem, printed slides 13-14.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Complete metric spaces - printed slide 9 to end of Chapter 1.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Complete metric spaces - printed slides 7-9.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Discussion of which topics from MTS will be most relevant in this module.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Complete metric spaces.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Introductory material on totally ordered sets and partially ordered sets.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Interlude on pointwise convergence and uniform convergence.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
printed slides 33-34 continued, completeness of the uniform norm.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
printed slides 33-34 continued, completeness of the uniform norm.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Lecture 10 - Normed spaces and Banach spaces - printed slides 31-34.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Normed spaces and Banach spaces - printed slides 28-31.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Infinite products and Tychonoff's theorem - printed slide 26 to end of chapter.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Discussion session on partially ordered sets - part b.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
The proof of Tychonoff's theorem.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Infinite products and Tychonoff's theorem.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Discussion session on partially ordered sets.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Chapter 2: Infinite products and Tychonoff’s theorem, printed slides 13-14.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Complete metric spaces - printed slide 9 to end of Chapter 1.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Complete metric spaces - printed slides 7-9.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Discussion of which topics from MTS will be most relevant in this module.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Complete metric spaces.
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951
Introductory material on totally ordered sets and partially ordered sets
Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951