Banach spaces and operator ideals (MAGIC088) |
GeneralDescription
We begin by introducing the concept of a Schauder basis for a Banach space. This is the natural analogue of an orthonormal basis for a Hilbert space, or a Hamel basis for a vector space; note, however, that in contrast to these examples, a Banach space may not have a Schauder basis. As an application of Schauder bases we shall prove that the ideal of compact operators is the only non-trivial closed ideal in the ring of all bounded linear operators on each of the classical sequence spaces lp (for 1 <= p < infinity) and c0; this result is due to Calkin (1941) for p=2 and to Gohberg, Markus, and Feldman (1960) in the general case.
SemesterSpring 2016 (Monday, January 11 to Friday, March 18) Timetable
PrerequisitesBasic knowledge of functional analysis, up to and including the Hahn-Banach Theorem; MAGIC061 covers this, and much more.
SyllabusSchauder bases in Banach spaces; Gohberg-Markus-Feldman's characterization of the closed ideals in the classical sequence spaces.
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Bibliography
Note: Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.) AssessmentDuring the official MAGIC 2-week exam period in April, there will be a single take-home exam. It will have three questions, each consisting of several parts. To pass, students must achieve a score of at least 50% overall. The examinable material is everything that we have covered in the lectures, corresponding roughly to Chapters 6-8 of the lecture notes. (You will not be expected to know the details of the proofs in Chapter 2, which we did not go through in detail, but you will need to know, and be able to apply, the standard results therein.)
The expectation is that the exam can be completed within a couple of hours, or a day at most if you want to typeset your answers (something which is good L^{A}T_{E}X training and much appreciated by the marker!)
Exam 2016 - MAGIC088 Banach spaces and operator ideals
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