MAST30026
Metric and Hilbert Spaces

Semester II 2015

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

Time and Location:
       Lecture: Tuesday 10:00 - 11:00 Richard Berry Russell Love Theatre
       Lecture: Wednesday 10:00 - 11:00 Richard Berry Russell Love Theatre
       Lecture: Thursday 10:00 - 11:00 Richard Berry Russell Love Theatre
       Practice class: Tuesday 16:15-17:15 Richard Berry Russell Love Theatre (no tutorial on 28 July)
       Practice class: Wednesday 9:00-10:00 Richard Berry Room 213 (no tutorial on 29 July)

Pre-Exam consultation hours are Tuesday 27 October at 10:00-11:00,
Thursday 29 October at 10:00-11:00, and
Wednesday 4 November at 10:00-11:00.
These will be held in Russell Love Theatre in Richard Berry.

The student representatives are
Khaya Mpehle email: Kmpehle@student.unimelb.edu.au and
Berooz Niknami email: bniknami@student.unimelb.edu.au

Announcements

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/2015/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 10: 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, Assignment 1 Question 8
Thursday, October 15: Assignment 2 (pdf file available HERE): Solutions Assignment 2 Question 1, Assignment 2 Questions 2 and 3, Assignment 2 Question 4.

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.

J. J. Koliha, Metrics, Norms and Integrals; An Introduction to Contemporary Analysis, World Scientific 2008
L Debnath and P. Mikusinski, Introduction to Hilbert spaces with applications, 2nd Edition, Academic Press, 1999
A. Bressan Lecture Notes on Functional Analysis American Mathematical Society, 2013.
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

29 July 2014 Lecture 1: Housekeeping and Proof Machine.
30 July 2014 Lecture 2: Holder, Minkowski and Cauchy-Schwarz inequalities; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
31 July 2014 Lecture 3: Metric spaces, normed vector spaces, l^p, L^p; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
1 August 2014 Lecture 4: Examples and HW questions; handwritten lecture notes (pdf file). Week 1 Vocab, questions, and examples from Anupama Pilbrow (page 1, page 2, page 3).
5 August 2014 Lecture 5: Topological spaces, interiors and closures; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
6 August 2014 Lecture 6: Continuous functions and connected sets; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
7 August 2014 Lecture 7: Connectedness and connected components; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
8 August 2014 Lecture 8: Connected in R are intervals; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file). Week 2 Vocab, questions, and examples from Anupama Pilbrow (pdf file).
12 August 2014 Lecture 9: Convergences, equivalent metrics, closure; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
13 August 2014 Lecture 10: Convergence, continuity and uniform continuity; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
14 August 2014 Lecture 11: Spaces of functions, uniform convergence; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
15 August 2014 Lecture 12: Examples and HW questions; Vocab, questions, and examples from Emma Kong (pdf file). Week 3 Vocab, questions, and examples from Anupama Pilbrow (pdf file).
19 August 2014 Lecture 13: Cauchy sequences and complete spaces; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
20 August 2014 Lecture 14: Examples of complete spaces; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
21 August 2014 Lecture 15: Completion of a metric space; handwritten lecture notes (pdf file). Vocab, questions, and examples from Emma Kong (pdf file).
22 August 2014 Lecture 16: Examples and HW questions; Week 4 Vocab, questions, and examples from Anupama Pilbrow (pdf file).
26 August 2014 Lecture 17: Compactness; handwritten lecture notes (pdf file).
27 August 2014 Lecture 18: Lecture 18: connected compact and the mean value theorem; handwritten lecture notes (pdf file).
28 August 2014 Lecture 19: Hausdorff, normal and path connected; handwritten lecture notes (pdf file).
29 August 2014 Lecture 20: Banach fixed point theorem; handwritten lecture notes (pdf file). Week 5 Vocab, questions, and examples from Anupama Pilbrow (pdf file).
2 September 2014 Baire's theorem -- Lecture given by Hyam Rubinstein
3 September 2014 Baire's theorem, second version -- Lecture given by Hyam Rubinstein
4 September 2014 Lecture 23: Banach spaces -- Lecture given by Hyam Rubinstein
5 September 2014 Lecture 24: Examples and HW questions; Week 6 Vocab, questions, and examples from Anupama Pilbrow (pdf file).
9 September 2014 Lecture 25: Construction of L¹ -- Lecture given by Hyam Rubinstein
10 September 2014 Lecture 26: Schauder bases -- Lecture given by Hyam Rubinstein
11 September 2014 Lecture 27: Compactness of the closed unit ball in finite dimensions and infinite dimensions -- Lecture given by Hyam Rubinstein
12 September 2014 Lecture 28: Examples and HW questions; Week 7 Vocab, questions, and examples from Anupama Pilbrow (pdf file).
16 September 2014 Lecture 29: Norms of linear operators handwritten lecture notes (pdf file).
17 September 2014 Lecture 30: Lecture 30: Examples of linear operators handwritten lecture notes (pdf file).
18 September 2014 Lecture 31: More examples of linear operators
19 September 2014 Lecture 32: Duals handwritten lecture notes (pdf file).
23 September 2014 Lecture 33: Inner product spaces and orthogonality; handwritten lecture notes (pdf file).
24 September 2014 Lecture 34: Hilbert spaces and orthogonal projections;
25 September 2014 Lecture 35: Lecture 35: Orthonormal sequences and Bessel's inequality; handwritten lecture notes (pdf file).
26 September 2014 Lecture 36: Lecture 36: Proof of Bessel's inequality and Hilbert space projections; handwritten lecture notes (pdf file).
7 October 2014 Lecture 37: Norms of self adjoint operators; handwritten lecture notes (pdf file).
8 October 2014 Lecture 38: More self adjoint operators;
9 October 2014 Lecture 39: Existence of eigenvectors for compact self adjoint operators;
10 October 2014 Lecture 40: Examples and HW questions;
11 October 2014 Lecture 41: Eigenspaces of self adjoint operators; handwritten lecture notes (pdf file).
12 October 2014 Lecture 42: Lecture 42: Bases of eigenvectors for compact self adjoint operators; handwritten lecture notes (pdf file).
13 October 2014 Lecture 43: Kinds of spaces and Cauchy-Schwarz review; handwritten lecture notes (pdf file).
14 October 2014 Lecture 44: Examples and HW questions;
18 October 2014 Lecture 45: Lecture 45: Product spaces and equivalent metrics; handwritten lecture notes (pdf file).
19 October 2014 Lecture 46: Lecture 46: Convergence; handwritten lecture notes (pdf file).
20 October 2014 Lecture 47: Lecture 47: Osmosis topics; handwritten lecture notes (pdf file).
21 October 2014 Lecture 48: Examples and HW questions;

Lectures this semester

Week 1: Proof machine, the zoo and osmosis topics (Tutorial sheet 1)

28 July 2015 Lecture 1: Housekeeping and Proof machine; handwritten lecture notes (pdf file)
Math Grammar: Definitions, Theorems and How to do Proofs (pdf file)
Examples of proofs written in proof machine (pdf file)
29 July 2015 Lecture 2: The zoo (topological spaces to normed vector spaces); handwritten lecture notes (pdf file).
30 July 2015 Lecture 3: Osmosis topics; handwritten lecture notes (pdf file).

Week 2: Examples, Cauchy-Schwarz and triangle inequalities (Tutorial 1 material) (Tutorial sheet 2)

4 August 2015 Lecture 4: $ℝ^{2}$ , Cauchy-Schwarz and triangle inequalities; handwritten lecture notes (pdf file).
5 August 2015 Lecture 5: Number systems that are metric spaces; handwritten lecture notes (pdf file).
6 August 2015 Lecture 6: $ℓ^{p}$ , Hölder and Minkowski inequalities; handwritten lecture notes (pdf file).

Week 3: New spaces from old, and comparing spaces (Tutorial sheet 3)

11 August 2015 Lecture 7: New spaces from old: product spaces, subspaces -- neighbourhoods and filters; handwritten lecture notes (pdf file).
12 August 2015 Lecture 8: interiors and closures, dense and nowhere dense sets; handwritten lecture notes (pdf file).
13 August 2015 Lecture 9: The Cantor set, bounded sets; handwritten lecture notes (pdf file).

Week 4: Open is not not closed and closed is not not open: No tutorial sheet this week, please focus on assignment

18 August 2015 Lecture 10: Baire's theorem on open dense sets; handwritten lecture notes (pdf file).
19 August 2015 Lecture 11: Connectedness and path connectedness; handwritten lecture notes (pdf file).
20 August 2015 Lecture 12: Continuity and continuity at a point; handwritten lecture notes (pdf file).

Week 5: Limits

25 August 2015 Lecture 13: Definitions of limits; handwritten lecture notes (pdf file).
26 August 2015 Lecture 14: Closures and limits; handwritten lecture notes (pdf file).
27 August 2015 Lecture 15: Continuity and limits; handwritten lecture notes (pdf file).

Week 6: Compactness (Tutorial sheet 5)

1 September 2015 Lecture 16: Cluster points and Compact spaces - the summary cartoon; handwritten lecture notes (pdf file).
2 September 2015 Lecture 17: For ℝ, closed is complete and bounded is ball compact; handwritten lecture notes (pdf file).
3 September 2015 Lecture 18: sequential compactness if and only if complete and totally bounded; handwritten lecture notes (pdf file).

Week 7: Completions (Tutorial sheet 6)

8 September 2015 Lecture 19: Pointwise and uniform convergence; handwritten lecture notes (pdf file).
9 September 2015 Lecture 20: Cauchy sequences, complete spaces and the construction of the completion; handwritten lecture notes (pdf file).
10 September 2015 Lecture 21: The completion is complete; handwritten lecture notes (pdf file).

Week 8: Applications

15 September 2015 Lecture 22: Connected subsets of ℝ are intervals, and the intermediate value theorem and the mean value theorem; handwritten lecture notes (pdf file).
16 September 2015 Lecture 23: contractive functions and the fixed point theorem; handwritten lecture notes (pdf file).
17 September 2015 Lecture 24: norm absolute convergence and completeness; handwritten lecture notes (pdf file).

Week 9: Bases and bounded linear operators

22 September 2015 Lecture 25: Hamel bases and Schauder bases; handwritten lecture notes (pdf file).
23 September 2015 Lecture 26: norms of linear operators; handwritten lecture notes (pdf file).
24 September 2015 Lecture 27: examples of linear operators; handwritten lecture notes (pdf file).

Week 10: Adjoints and orthogonality (Tutorial 6 October 2015)

6 October 2015 Lecture 28: adjoints, dual bases and transpose; handwritten lecture notes (pdf file).
7 October 2015 Lecture 29: inner product spaces, orthogonality and Gram-Schmidt; handwritten lecture notes (pdf file).
8 October 2015 Lecture 30: Hilbert spaces, orthogonal projections and Bessel's inequality; handwritten lecture notes (pdf file).

Week 11: Eigenvalues (Tutorial 13 October 2015)

13 October 2015 Lecture 31: Existence of eigenvectors for compact self adjoint operators; handwritten lecture notes (pdf file).
14 October 2015 Lecture 32: inner product spaces, orthogonality and Gram-Schmidt; handwritten lecture notes (pdf file).
15 October 2015 Lecture 33: Hilbert spaces, orthogonal projections and Bessel's inequality; handwritten lecture notes (pdf file).

Week 12: Bessel's inequality and orthogonal decomposition

20 October 2015 Lecture 34: Bessel's inequality; handwritten lecture notes (pdf file).
21 October 2015 Lecture 35: Orthogonal decomposition; handwritten lecture notes (pdf file).
22 October 2015 Lecture 36: Quantum Mechanics; handwritten lecture notes (pdf file).