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MAST30026
Metric and Hilbert Spaces

Semester II 2017

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

Time and Location:
       Lecture: Tuesday 13:00 - 14:00 Peter Hall building Russell Love Theatre
       Lecture: Thursday 14:15 - 15:15 Peter Hall building Russell Love Theatre
       Lecture: Friday 13:00 - 14:00 Peter Hall building Russell Love Theatre
       Practice class: Tuesday 16:15-17:15 Peter Hall building Room 213
       Practice class: Thursday 12:00-13:00 Peter Hall building Room 213

Pre-Exam consultation hours of Arun Ram are Tuesday 24 October 3-5pm, Thursday 26 October 2-3pm and Monday 30 October 2-4pm.
These will be held in Room 174 in Richard Berry.

Combined sample exams handed out in tutorials (pdf file available HERE).

Arun Ram's consultation hours are Tuesdays 14:00-16:15, and Thursdays 13:00-14:15 in Room 174 of Richard Berry.

The student representatives are
Quentin Bell   email: qbell@student.unimelb.edu.au   and
Daniel Johnston   email: johnstond1@student.unimelb.edu.au

Announcements

  • 07.09.2017 The second assignment (due 12 October) is available (pdf file available HERE)
  • Jon Xu will provide additional consultation hours before the first assignment is due:
    11am-12pm Friday 1 Sept in Room G38 Peter Hall Building
    1:30pm-3:30pm Monday 4 Sept Thomas Cherry Room Peter Hall Building
    2pm-3pm Wednesday 6 Sept Thomas Cherry Room Peter Hall Building
  • it is required to complete the plagiarism declaration in LMS, by clicking on the Plagiarism declaration link for this subject and completing the submission of the Plagiarism declaration through LMS for this subject.
  • 26.07.2017 The first assignment (due 7 September) is available (pdf file available HERE)
  • The lectures will not be recorded.
  • Prof. Ram reads email but generally does not respond by email.
  • The start of semester pack includes: Housekeeping (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/2017/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 2pm on the following dates:

  • Thursday, Sep 7: 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, Assignment 1 Question 9,
  • Thursday, October 12: Assignment 2 (pdf file available HERE):   Solutions Assignment 2 Question 1, Assignment 2 Question 2, Assignment 2 Question 3, Assignment 2 Question 5, Assignment 2 Question 6, Assignment 2 Question 7,
If you cannot get this assignment, or any other class materials from this site, please ask the lecturer for a printed copy.

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.

  • it is required to complete the plagiarism declaration in LMS, by clicking on the Plagiarism declaration link for this subject and completing the submission of the Plagiarism declaration through LMS for this subject.
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 and Problem lists by Arun Ram (together in a single bound set) will be available for sale in the bookroom.

Additional References

The following additional references are recommended.

  • (This is in the High Use section of the ERC library) A. Bressan, Lecture Notes on Functional Analysis, American Mathematical Society, 2013.
  • (Not sure if Melbourne Univ. Library has this any more) S. Lang, Real analysis, Addison-Wesley, 1983
  • (This is available online through Melbourne University library) T. Tao, Analysis, Hindustan Book Agency, 2009 (2 volumes).

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 from Semester II 2014 HERE

Lectures from Semester II 2015 HERE

Lectures from Semester II 2016 HERE

Lectures this semester

  • Week 1:
    • 25 July 2016 Lecture 1: Limits: Compactness; handwritten lecture notes (pdf file)
    • 27 July 2016 Lecture 2: Spaces: The Zoo; handwritten lecture notes (pdf file).
    • 28 July 2016 Lecture 3: Osmosis topics: Proof machine;
      Math Grammar: Definitions, Theorems and How to do Proofs (pdf file)
      handwritten lecture notes (pdf file).

  • Week 2: Tutorial sheet 1
    • 1 August 2016 Lecture 4: Limits: Sequences and Completeness; handwritten lecture notes (pdf file).
    • 3 August 2016 Lecture 5: Spaces: Neighbourhoods, Continuous and Uniformly Continuous functions; handwritten lecture notes (pdf file).
    • 4 August 2016 Lecture 6: Osmosis topics: Partial orders, least upperbounds, interiors and closures; handwritten lecture notes (pdf file).

    • Tutorial: Learning proof machine via the B(V,W) is complete proof.

  • Week 3:
    • 9 August 2016 Lecture 7: Limits: Cluster points and limit points; handwritten lecture notes (pdf file).
    • 10 August 2016 Lecture 8: Spaces: Limits in Topological Spaces; handwritten lecture notes (pdf file).
    • 11 August 2016 Lecture 9: Osmosis topics: Real numbers, p-adic numbers and polynomials; handwritten lecture notes (pdf file).

    • Tutorial: Learning exam writing by doing exam marking.

  • Week 4:
    • 16 August 2016 Lecture 10: Cover compactness, ball compactness and bounded sets; handwritten lecture notes (pdf file).
    • 17 August 2016 Lecture 11: Spaces: Subspaces, product topologies and product metrics; handwritten lecture notes (pdf file).
    • 18 August 2016 Lecture 12: Osmosis topics: Real numbers -- the order, the metric, Archimides' property and the least upper bound property; handwritten lecture notes (pdf file).

    • Tutorial: Outline/List of the course topics via the sample solutions in Part III of the notes.

  • Week 5:
    • 23 August 2016 Lecture 13: Limits, closure and Hausdorff spaces; handwritten lecture notes (pdf file).
    • 24 August 2016 Lecture 14: Spaces: Topological equivalence; handwritten lecture notes (pdf file).
    • 25 August 2016 Lecture 15: Osmosis topics: Topology of the real numbers and connectedness; handwritten lecture notes (pdf file).

    • Tutorial: Outline/List of the topics and EXAMPLES via the questions in Part II Chapter 3 and Chapter 6 of the notes.

  • Week 6: Tutorial sheet Week 6
    • 30 August 2016 Lecture 16: Limits: Nowhere Dense sets and normed vector spaces; handwritten lecture notes (pdf file).
    • 31 August 2016 Lecture 17: Spaces: Function spaces; handwritten lecture notes (pdf file).
    • 1 September 2016 Lecture 18: Osmosis topics: Inner product spaces and Cauchy-Schwarz; handwritten lecture notes (pdf file).

    • Tutorial: Questions towards completion of the assignment.

  • Week 7: Vector spaces and linear transformations
    • 5 September 2016 Lecture 19: Absolute Convergence in normed vector spaces; handwritten lecture notes (pdf file).
    • 7 September 2016 Lecture 20: Bounded linear transformations; handwritten lecture notes (pdf file).
    • 8 September 2016 Lecture 21: Osmosis topics: Vector spaces and bases; handwritten lecture notes (pdf file).

  • Week 8: Projections
    • 12 September 2016 Lecture 22: Projections onto closed subspaces; handwritten lecture notes (pdf file).
    • 14 September 2016 Lecture 23: Duals and adjoints and the Riesz Representation Theorem; handwritten lecture notes (pdf file).
    • 15 September 2016 Lecture 24: Orthonormal sequences; handwritten lecture notes (pdf file).

  • Week 9: Eigenvectors and eigenvalues
    • 19 September 2016 Lecture 25: Eigenvalues; handwritten lecture notes (pdf file).
    • 21 September 2016 Lecture 26: Existence of Eigenvectors; handwritten lecture notes (pdf file).
    • 22 September 2016 Lecture 27: Properties of eigenspaces; handwritten lecture notes (pdf file).

  • Week 10:
    • 3 October 2016 Lecture 28: The Spectral Theorem; handwritten lecture notes (pdf file).
    • 5 October 2016 Lecture 29: Examples of Linear operators; handwritten lecture notes (pdf file).
    • 6 October 2016 Lecture 30: Adjoints and the Riesz Representation Theorem; handwritten lecture notes (pdf file).

  • Week 11:
    • 11 October 2016 Lecture 31: Completions; handwritten lecture notes (pdf file).
    • 12 October 2016 Lecture 32: Matrix norms; handwritten lecture notes (pdf file).
    • 13 October 2016 Lecture 33: Computation, norms and writing; handwritten lecture notes (pdf file).

  • Week 12:
    • 18 October 2016 Lecture 34: p-adic numbers; handwritten lecture notes (pdf file).
    • 19 October 2016 Lecture 35: Filter, Hausdorff spaces and compact Spaces: handwritten lecture notes (pdf file).
    • 20 October 2016 Lecture 36: Last Lecture.