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MAST30026
Metric and Hilbert Spaces

Semester II 2014

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: Friday 10:00-11:00 Richard Berry Russell Love Theatre

Consultation hours are Wednesdays 11:00-13:00 and Fridays 11:00-12:00 in Room 174 of Richard Berry. Consultation hours will not be held during the weeks of 4 August, 1 September and 8 September.

Announcements

  • Prof. Ram reads email but generally does not respond.
  • The start of semester pack includes: Housekeeping (pdf file), Beyond Third Year (pdf file), Vacation Scholar Flyer (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/2014/MAST30026.

This subject extends ideas and results about limits and continuity from Euclidean spaces to very general situations, for example spaces of functions and manifolds. 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. The material is important throughout analysis, geometry and topology, and has applications to numerical mathematics, differential and integral equations, optimisation, physics, logic, computing and algebra.

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:

  • Wednesday, Sep 10: Assignment 1 (or pdf file available HERE):   Assignment 1 solutions (or pdf file of Questions 1-4 available HERE   and pdf file of Assignment 1 solutions Questions 5-10 available HERE)
  • Wednesday, October 15: Assignment 2 (or pdf file available HERE):   Assignment 2 solutions (or pdf file of Questions 1-4 available HERE   and pdf file of Assignment 2 solutions Questions 5-8 available HERE)
Assignments will be handed out in lectures approximately one week 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 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.

Vocabulary

adherent point boundary of a set bounded function bounded set Cantor set
Cauchy Schwarz inequality closed ball closed set continuous at a point continuous function
convergent sequence dense set discrete metric discrete space distance between sets
distance between point and set equivalent metrics Euclidean metric Euclidean space Holder inequality
interior point interior of a set isolated point limit of a sequence metric
metric space metric subspace Minkowski inequality neighbourhood of a point norm
normed vector space nowhere dense set open ball open set pointwise convergent
product metric space standard metric standard metric subsequence triangle inequality
uniformly continuous function uniformly convergent almost everywhere equal functions bijective function
Cauchy sequence compact space complete set complete space completion
continuous function contraction convergent series C(X) fixed point
full set homeomorphism injective function isometric spaces isometry
norm-absolutely convergent series null set step function surjective function topological space
B(X,Y) Banach space Bessell's inequality
bounded linear operator connected space connected set connected component cover
direct sum disconnected space epsilon net finite intersection property Fourier coefficients
Fourier series Gram-Schmidt process Hausdorff space Heine-Borel property Hilbert space
inner product interval invertible linear operator l2 lp
L2(X) linear functional normal space open cover operator norm
orthogonal complement orthonormal orthonormal basis path path connected
Schauder basis separable space separation subcover total set
totally bounded adjoint compact linear operator complement
complex numbers dual space eigenspace eigenvector empty set
equicontinuous family Fredholm integral operator inf integers inverse function
isometry natural numbers positive linear operator rational numbers real numbers
Riesz representation theorem Spectral expansion theorem subset sup 2-valued function
self adjoint unitary operator unit circle unit sphere

References

The following additional references will be on reserve in the Mathematics 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.

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

  • 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, lp, Lp; 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;