MAST30026
Metric and Hilbert Spaces

Semester II 2016

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

Time and Location:
       Lecture: Tuesday-Wednesday-Thursday 10:00 - 11:00 Richard Berry Russell Love Theatre
       Practice class: Tuesday 16:15-17:15 Richard Berry Russell Love Theatre
       Practice class: Wednesday 9:00-10:00 Richard Berry Room 213

Pre-Exam consultation hours of Arun Ram are Tuesday 1 November 9:00-12:00
These will be held in Room 174 in Richard Berry.

Arun Ram's consultation hours are Tuesdays 9:00-10:00, Wednesdays 11:00-12:00, and Thursdays 11:00-12:00 in Room 174 of Richard Berry.

Anupama Pilbrow's consultation hours are Tuesdays 11:00-12:00, Wednesday 2:00-3:00, and Thursday 2:00-3:00 in Room G13 Richard Berry

The student representatives are
Adalya Nash email: adalyan@student.unimelb.edu.au and
Chee Yeung Chun email: cheec1@student.unimelb.edu.au

Announcements

29.09.2016 Slight updates to Question 7 on the second assignment (due 13 October) to make it slightly easier (pdf file available HERE)
21.08.2016 The second assignment (due 13 October) is now available (pdf file available HERE)
31.07.2016 The first assignment (due 8 September) is now available (pdf file available HERE)
The lectures will not be recorded.
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).

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2016/MAST30026.

This subject provides a basis for further studies in modern analysis, geometry, topology, differential equations and quantum mechanics.It introduces the idea of a metric space with a general distance function, and the resulting concepts of convergence, continuity, completeness, compactness and connectedness. The subject also introduces Hilbert spaces: infinite dimensional vector spaces (typically function spaces) equipped with an inner product that allows geometric ideas to be used to study these spaces and linear maps between them.

Topics include: metric and normed spaces, limits of sequences, open and closed sets, continuity, topological properties, compactness, connectedness; Cauchy sequences, completeness, contraction mapping theorem; Hilbert spaces, orthonormal systems, bounded linear operators and functionals, applications.

Assessment

There will be one three hour examination at the end of the semester, and two written assignments during semester. For your final mark, the exam counts for 80% and the assignments count for a total of 20% (10% each). Note that each piece of assessment is compulsory.

Assignments

Assignments will be due by 10am on the following dates:

Thursday, Sep 8: Assignment 1 (pdf file available HERE): Solutions Assignment 1 Question 1, Assignment 1 Question 2, Assignment 1 Question 3, Assignment 1 Question 4, Assignment 1 Question 5, Assignment 1 Question 6, Assignment 1 Question 7,
Thursday, October 13: Assignment 2 (pdf file available HERE): Solutions Assignment 2 Question 1, Assignment 2 Question 2, Assignment 2 Question 3, Assignment 2 Question 4, Assignment 2 Question 5, Assignment 2 Question 6, Assignment 2 Question 7.

Assignments will be handed out in lectures approximately four weeks before the due date. Copies will also be available through the 30026 web site.

These assignments must be your own work. While students are encouraged to discuss their coursework and problems with one another, assignments must be written up independently. It is University policy that students submit a signed plagiarism sheet at the start of each semester. If you do not submit this sheet your assignments will be given a mark of zero.

The plagiarism declaration is available here.

Students who are unable to submit an assignment on time and qualify for special consideration should contact the lecturer as soon as possible after the due date.

Prerequisites

Group theory and linear algebra and one of
Real analysis with applications or Accelerated mathematics 2.

Lecture notes

Lecture notes by Prof. J. Hyam Rubinstein and Arun Ram (together in a single bound set) will be available for sale in the bookroom.

Problem sheets

HW questions to work on distilled from lecture notes by Prof. J. Hyam Rubinstein.

Problem sheets from a previous semester prepared by Prof. J. Hyam Rubinstein.

ALSO look at Chapter 10 of the "Little book on convergence" by Arun Ram, included as the second part of the notes that you purchased from the bookroom.

References

The following additional references will be on reserve in the ERC Library.

A. Bressan Lecture Notes on Functional Analysis American Mathematical Society, 2013.
S. Lang, Real anlysis, Addison-Wesley, ????
W. Rudin, Principles of mathematical analysis, McGraw Hill, .
T. Tao, Analysis, Hindustan Book Agency, 2009.

Recommended links from Arun Ram: Notes :

Lectures from Semester II 2014 HERE

Lectures from Semester II 2015 HERE

Lectures this semester

Week 1: Topology, Continuous functions and Metric spaces (Chapter 2 in the Notes)

26 July 2016 Lecture 1: Topologies and uniformities handwritten lecture notes (pdf file)
27 July 2016 Lecture 2: Continuous and uniformly continuous functions; handwritten lecture notes (pdf file).
28 July 2016 Lecture 3: Metric spaces, metric space topology and metric space uniformity; handwritten lecture notes (pdf file).

Week 2: Examples (Chapter 4 in the notes) (Tutorial sheet 1)

2 August 2016 Lecture 4: Proof machine; Math Grammar: Definitions, Theorems and How to do Proofs (pdf file)
Examples of proofs written in proof machine (pdf file)
3 August 2016 Lecture 5: Number systems that are metric spaces; handwritten lecture notes (pdf file).
4 August 2016 Lecture 6: Examples of polynomial number systems and function spaces handwritten lecture notes (pdf file).

Week 3: New spaces from old (Chapter 5 in the notes) (Tutorial sheet 2)

9 August 2016 Lecture 7: Bounded sets, bounded functions and bounded linear operators; handwritten lecture notes (pdf file).
10 August 2016 Lecture 8: Inequalities: Cauchy-Schwarz, the triangle inequality, Höder and Minkowski inequalities; handwritten lecture notes (pdf file).
11 August 2016 Lecture 9: Subspaces and products; handwritten lecture notes (pdf file).

Week 4: Neighborhoods, close points and limits (Chapter 3 in the Notes) (Tutorial sheet 3)

16 August 2016 Lecture 10: Interiors and closures; handwritten lecture notes (pdf file).
17 August 2016 Lecture 11: Limits and closures; handwritten lecture notes (pdf file).
18 August 2016 Lecture 12: Hausdorff spaces; handwritten lecture notes (pdf file).

Week 5: Compact spaces (Chapter 6 in the notes) (Tutorial sheet 4)

23 August 2016 Lecture 13: Initial results and Counterexamples; handwritten lecture notes (pdf file).
24 August 2016 Lecture 14: Cover compact implies Ball compact implies bounded and Sequentially compact implies Cauchy compact implies closed; handwritten lecture notes (pdf file).
25 August 2016 Lecture 15: Cover compact if and only if sequentially compact if and only if ball compact and Cauchy compact; handwritten lecture notes (pdf file).

Week 6: Connectedness, Uniform and pointwise convergence, and nowhere dense sets (Tutorial sheet 5)

30 August 2016 Lecture 16: Connected components; handwritten lecture notes (pdf file).
31 August 2016 Lecture 17: Pointwise and uniform convergence and Cantor's set; handwritten lecture notes (pdf file).
1 September 2016 Lecture 18: Nowhere dense sets and Baire's theorem; handwritten lecture notes (pdf file).

Week 7: Completions, Fixed point theorem and Absolute convergence

6 September 2016 Lecture 19: Locally compact spaces and Completions; handwritten lecture notes (pdf file).
7 September 2016 Lecture 20: Contraction mappings and the Banach fixed point theorem; handwritten lecture notes (pdf file).
8 September 2016 Lecture 21: Absolute convergence and completeness; handwritten lecture notes (pdf file).

Week 8: Bases and Linear operators

13 September 2016 Lecture 22: Hamel and Schauder bases; handwritten lecture notes (pdf file).
14 September 2016 Lecture 23: Bounded linear operators; handwritten lecture notes (pdf file).
15 September 2016 Lecture 24: Orthonormal sequences and Gram-Schmidt; handwritten lecture notes (pdf file).

Week 9: Projections and eigenvectors (First year Tutorial sheet also applicable here)

20 September 2016 Lecture 25: Projections and orthogonal decomposition; handwritten lecture notes (pdf file).
21 September 2016 Lecture 26: Riesz representation theorem; handwritten lecture notes (pdf file).
22 September 2016 Lecture 27: Bessel's inequality and Fourier decomposition; handwritten lecture notes (pdf file).

Week 10: Adjoints and orthogonality

4 October 2016 Lecture 28: Types of Linear Operators; handwritten lecture notes (pdf file).
5 October 2016 Lecture 29: Eigenspaces of Linear Operators; handwritten lecture notes (pdf file).
6 October 2016 Lecture 30: Eigenvectors and power iteration; handwritten lecture notes (pdf file).

Week 11: Eigenvalues

11 October 2016 Lecture 31: Norms of Matrix Linear Operators; handwritten lecture notes (pdf file).
12 October 2016 Lecture 32: The dual of ℓ^p; handwritten lecture notes (pdf file).
13 October 2016 Lecture 33: Examples of linear operators; handwritten lecture notes (pdf file).

Week 12: Bessel's inequality and orthogonal decomposition (Tutorial sheet -- Final week)

18 October 2016 Lecture 34: Bounded, continuous and uniformly continuous Operators; handwritten lecture notes (pdf file).
19 October 2016 Lecture 35: Limits, Neighborhoods, Uniform spaces, Cauchy sequences and uniform confinuity; handwritten lecture notes (pdf file).
20 October 2016 Lecture 36: Last Lecture.