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MAST90020
Functional Analysis

Semester II 2014

Lecturer: Arun Ram, 174 Richard Berry, email: aram@unimelb.edu.au

Time and Location:
       Lecture: Monday 13:00 - 14:00 Room 213 Richard Berry
       Lecture: Wednesday 13:00 - 14:00 Room 213 Richard Berry
       Lecture/Practical Friday 14:15-15:15 Room 213 Richard Berry

Announcements

  • No books, notes, calculators, ipods, ipads, phones, etc at the exam.
  • Consultation hours are Wednesdays 11:00-13:00 and Friday 11:00-12:00 in Room 174 Richard Berry.
  • Prof. Ram reads email but generally does not respond.
  • The start of semester pack includes: Housekeeping (pdf file), Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), SSLC responsibilities (pdf file).
  • It is University Policy that:           

    “a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results.  Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).

    Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).
  • Printing arrangements from computer Lab G70: Students must use their Unicards to print documents. Locations for Unicard uploaders can be found at: http://www.studentit.unimelb.edu.au/images/facilities/autoloaders.gif

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2014/MAST90020. We will use the book by Alberto Bressan,

A. Bressan, Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations, Graduate Studies in Mathematics Vol. 143, American Mathematical Society, 2013. ISBN-10: 0-8218-8771-8 ISBN-13: 978-0-8218-8771-4
The table of contents lists the following chapters:

  • Background material
  • Banach spaces
  • Spaces of continuous functions
  • Bounded linear operators
  • Hilbert spaces
  • Compact operators on a Hilbert space
  • Semigroups of linear operators
  • Sobolev spaces
  • Linear partial differential equations
and this is the material that we will cover in the course.

Assessment

There will be one homework assignment per week (including the first week of class, but not the last week of class, for eleven assignments total). Each homework assignments will consist of one problem from the book -- each student will be assigned a different problem. Writing clarity and proof style will be the primary factor in assessment of the homework assignments. Accuracy in the use of proof machine will be required. (Math Grammar: Definitions, Theorems and How to do Proofs (pdf file) and Examples of proofs written in proof machine (pdf file))

The homework assignments will be marked at 10 marks each. In addition to the weekly assignments (110/210=52%, eleven assignments worth just under 5% each), there will be a three-hour written examination (100/210=48%, in the examination period).

The plagiarism declaration is available here.

  • The final exam will be 3 hours.

Texts

  • Prescribed text: A. Bressan, Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations, Graduate Studies in Mathematics Vol. 143, American Mathematical Society, 2013. ISBN-10: 0-8218-8771-8 ISBN-13: 978-0-8218-8771-4

  • Recommended links from Arun Ram: Notes :
    • Banach and Hilbert Spaces
    • Topological spaces and continuous functions, Interiors and closures
    • Filters, limits and continuous functions, limits and continuous functions proofs, Examples in R and C: Limits, and Sequences, and Series
    • Uniform spaces, metric spaces and completion, completion in the h-adic topology
    • Hausdorff and separable spaces
    • Compact spaces and proper mappings
    • Irreducible and Noetherian topological spaces
    • Measurable spaces, measurable sets and measurable functions
    • Measures and Integration
    • Lebesgue convergence theorems
    • Function spaces
    • The Radon-Nikodym and Reisz representation theorems
    • Distributions: The Riesz representation theorem

Lectures

  • 28 July 2014: Housekeeping, Proof machine; handwritten lecture notes (pdf file).
  • 30 July 2014: Lecture 2: Bressan Chapter 1; handwritten lecture notes (pdf file).
  • 1 August 2014: Lecture of Vince Vatter (University of Florida).
  • 4 August 2014: Bressan Chapter 2; handwritten lecture notes (pdf file).
  • 6 August 2014: More Bressan Chapter 2; handwritten lecture notes (pdf file).
  • 8 August 2014: Even more Bressan Chapter 2; handwritten lecture notes (pdf file).
  • 11 August 2014: Bressan Chapter 3; handwritten lecture notes (pdf file).
  • 13 August 2014: Bressan Chapter 3; handwritten lecture notes (pdf file).
  • 15 August 2014: More Bressan Chapter 3;
  • 18 August 2014: Bressan Chapter 4;
  • 20 August 2014: Bressan Chapter 4;
  • 22 August 2014: More Bressan Chapter 4;
  • 25 August 2014: Bressan Chapter 5; handwritten lecture notes (pdf file).
  • 27 August 2014: Bressan Chapter 5; handwritten lecture notes (pdf file).
  • 29 August 2014: More Bressan Chapter 5;
  • 1 September 2014: Bressan Chapter 6; handwritten lecture notes (pdf file).
  • 3 September 2014: Bressan Chapter 6;
  • 5 September 2014: More Bressan Chapter 6;
  • 8 September 2014: Bressan Chapter 7; handwritten lecture notes (pdf file).
  • 10 September 2014: Bressan Chapter 7;
  • 12 September 2014: More Bressan Chapter 7;
  • 15 September 2014: Bressan Chapter 8;
  • 17 September 2014: Bressan Chapter 8;
  • 19 September 2014: More Bressan Chapter 8;
  • 22 September 2014: Bressan Chapter 9;
  • 24 September 2014: Bressan Chapter 9;
  • 26 September 2014: More Bressan Chapter 9;
  • 6 October 2014: Bressan Chapter 10;
  • 8 October 2014: Bressan Chapter 10;
  • 10 October 2014: More Bressan Chapter 10;
  • 13 October 2014: Bressan Chapter 11;
  • 15 October 2014: Bressan Chapter 11;
  • 17 October 2014: More Bressan Chapter 11;
  • 20 October 2014: Bressan Chapter 12;
  • 22 October 2014: Bressan Chapter 12;
  • 24 October 2014: More Bressan Chapter 12;