
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 10:0011:00 Richard Berry Russell Love Theatre
Consultation hours are Wednesdays 11:0013:00 and Fridays 11:0012:00 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 ‘makeup’ 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:
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

normabsolutely 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

GramSchmidt process

Hausdorff space

HeineBorel property

Hilbert space

inner product

interval

invertible linear operator

l^{2}

l^{p}

L^{2}(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

2valued 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 :
Lectures
 29 July 2014
Lecture 1:
Housekeeping and Proof Machine.
 30 July 2014
Lecture 2:
Holder, Minkowski and CauchySchwarz 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 L^{1}  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 CauchySchwarz 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;