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General


Description

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.

Semester

Spring 2016 (Monday, January 11 to Friday, March 18)

Timetable

  • Fri 11:05 - 11:55

Prerequisites

Basic knowledge of functional analysis, up to and including the Hahn-Banach Theorem; MAGIC061 covers this, and much more.

Syllabus

Schauder bases in Banach spaces; Gohberg-Markus-Feldman's characterization of the closed ideals in the classical sequence spaces.

Lecturer


Niels Laustsen
Email n.laustsen@lancaster.ac.uk
Phone (01524) 594617
Interests Operators on Banach spaces, Banach algebras, K-theory
Photo of Niels Laustsen


Students


Photo of Amos Ajibo
Amos Ajibo
(Newcastle)
Photo of Sean Dewar
Sean Dewar
(Lancaster)
Photo of Jason Hancox
Jason Hancox
(Lancaster)
Photo of Bence Horvath
Bence Horvath
(Lancaster)
Photo of Tanmay Inamdar
Tanmay Inamdar
(East Anglia)
Photo of Mateusz Jurczynski
Mateusz Jurczynski
(Lancaster)
Photo of Christopher Menez
Christopher Menez
(Lancaster)
Photo of Richard Skillicorn
Richard Skillicorn
(Lancaster)
Photo of ANON STUDENT
ANON STUDENT
(*External)
Photo of Batzorig Undrakh
Batzorig Undrakh
(Newcastle)
Photo of CELLAN WHITE
CELLAN WHITE
(Cardiff)
Photo of Jared White
Jared White
(Lancaster)


Bibliography


Lecture notesNiels Laustsen
A Short Course on Banach Space TheoryCarothers
An Introduction to Banach Space TheoryMegginson


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.)

Assessment



During 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 LATEX training and much appreciated by the marker!)

Exam 2016 - MAGIC088 Banach spaces and operator ideals

Files:Exam paper
Released: Monday 18 April 2016 (523.6 days ago)
Deadline: Friday 29 April 2016 (511.6 days ago)
Instructions:

Attempt all questions. The total number of marks is 50, broken down as indicated in the right-hand margin of the exam paper. The minimum pass mark is 25. To obtain full marks, you must give details of your workings and include clear references (by quoting the relevant theorem numbers) to any results from the lecture notes that you use. You are not expected to reprove any results that have been proved in the course. If you use sources other than the course materials, you must give clear reference to them in your answers.



Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
exam16.pdf
slides11.pdf
slides12.pdf
slides13.pdf
slides14.pdf
slides15.pdf
slides16.pdf
slides17.pdf
slides18.pdf
slides19.pdf
slides20.pdf
solutionsexam16.pdf
10-25lecturenotes.pdf


Recorded Lectures


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