
MAST30026
Metric and Hilbert Spaces

Semester II 2022 
PreExam consultation hours of Arun Ram are Monday 31 October 1:302:30pm,
Wednesday 2 November 1:302:30pm and Friday 4 November 1:302:30pm.
These will be held in Room G47 in the Peter Hall building.
Proof writing sessions
The exam assesses whether you can write quality
solutions to questions in an exam setting. So the main goal of
the class is to learn to write quality, well presented solutions,
that communicate well and thoroughly to the reader. Whether or not
you get a correct answer has much less importance than
whether your exposition is of good quality.
To help learn this skill, we will provide three "proof writing" sessions
a week, in the same room as the lecture, in the hour before class.
These are not required,
you decide whether you want to take advantage of
this resource or not. It is during these sessions that we will provide models
for writing solutions to exam questions.
These sessions will be available live streamed and recorded, via
the usual Lecture capture.
https://canvas.lms.unimelb.edu.au/courses/129807
The goal is to be able to write quality solutions in an exam setting.
Lectures
Lecturer: Arun Ram, 174 Peter Hall, email: aram@unimelb.edu.au
Time and Location: Zoom link on Canvas
https://canvas.lms.unimelb.edu.au/courses/129807
Lecture: Monday 9:0010:00 Russell Love Theatre, Peter Hall Building,
Lecture: Wednesday 9:0010:00 Russell Love Theatre, Peter Hall Building,
Lecture: Friday 10:0011:00 Russell Love Theatre, Peter Hall Building,
The lectures will be live streamed by the usual Echo 360/Canvas delivery
If there are no technology glitches, each lecture will also be Zoom recorded
and made available on Echo360 (accessible through Canvas) within 7 days
after the live lecture.
You will do well on the exam if you are able to write solutions in the same
model as in the assignment solutions and the proof writing sessions
without the need for notes.
This is the skill that will be assessed on the exam.
Tutorials
The tutorials will begin on Friday of Week 1 (July 29).
In the first tutorial the "assignment marking exercise groups" will
be allocated. If you do not attend the first tutorial the tutor will assign
you to a group without your input.
Practice class: Monday 15:1516:15 Online,
Tutor: Weiying Guo
Practice class: Wednesday 11:0012:00 in Peter Hall 213,
Tutor: Arun Ram
Practice class: Friday 11:0012:00 in Peter Hall 213,
Tutor: Arun Ram
Discussion
A key part of Lecture time is the Ask me a question time.
Given that we will hold the "Proof writing workshop" in the hour before lecture
these will probably morph into the "Ask me a question" time for the 10 minute
slot between class periods. The official lecture starts at 5 min
past the hour.
Consultation hours: Arun Ram, MondayWednesday 1011am,
Student Representatives: Haris Rao murao@student.unimelb.edu.au
and Jiani Xie jianix1@student.unimelb.edu.au
Facebook group:
https://www.facebook.com/groups/????????/
The lecturer and tutors will not be reading, looking at, or participating
in the Facebook group. If you wish to have the lecturer or tutors
read or participate, hold the discussion on Ed Discussion
(which has math equation/LaTeX capability).
The Ed Discussion link for this course
is available through Canvas.
https://canvas.lms.unimelb.edu.au/courses/129807
Methods and mechanics for teaching, and how to improve teaching
and learning are topics that I think about quite alot. One piece of
writing that I have done on the subject is
Teaching Math in The Next Life
which might also be helpful to students taking this course. Certainly
we will discuss many of these topics in class. I would be very grateful
for any reactions/thoughts/discussion that you can give me on this
writing (and any of the other resources on this page).
Assignments
The homework assignments are at the following links:
(total 7% each = 5% for the first submission +2% for the second submission)
 Assignment 1: Due 11 and 18 August before 4:00pm.
Solutions:
Question 1,
Question 2,
Question 3,
Question 4A,
Question 4B,
Question 4D,
Question 4E,
Question 4F.
 Assignment 2 (question 5(b) revised 27.08.2002): Due 8 and 15 September before 4:00pm.
Solutions:
Assignment 2 solutions,
 Assignment 3: Due 6 and 13 October before 4:00pm,
Solutions:
Question 1,
Question 2,
Question 3,
Question 4,
Question 5.
For each assignment submit your assigment solutions through
Canvas/Gradescope before the first deadline
and submit the marking exercise before the second deadline.
The first deadline is for you to submit your solutions to
the assignment questions.
Your submission will be marked by the tutors. The first submission
of each assignment is worth 5% of the total mark for the course.
After the first submission you will receive 4 assignments to 'mark'
(the submissions of the others in you 'assignment marking group').
Only by reading solutions and trying to do marking yourself,
do you learn how to optimise your marks on the exam.
Read the solutions, provide feedback, and assign a mark for each question.
You may be asked to provide justification for the mark you assign
(so keep records of the marking scheme you used for each question).
Submit your "marking" of the 4 assignments as
the second submission of the assignment.
Tutors will allocate marks for your second submission
based on how conscientiously you undertake this exercise.
The second submission of each assignment is worth 2% of
the total mark for the course.
Because the assignment solutions will be released immediately after
the due date/time, no late submissions will be accepted.
If you miss a submission and you have a valid reason
(medical certificate or equivalent), then you will be given the option
to recover the marks for that assignment
by making 3 distinct handwritten copies of the
assignment solutions and one
handwritten copy of the sample exam solutions.
It is recommended that you do and submit the assignments in
the same way that you will for the exam: handwritten, a new page
for each question, very clear exposition so that the marker can follow
your argument easily, scanned with Camscanner, and uploaded
question by question into Gradescope.
One purpose of the assignments is to prepare your exam taking skills.
Assessment
Assessment will be based on
three written assignments due at regular intervals during semester
amounting to a total of up to 50 pages (20%),
and a 3hour written examination in the examination period (80%).
Almost everybody seems to agree that the "50 pages" doesn't make much sense
in our modern technological world where some assignments might even be submitted
in video form (the handbook entry has not been changed for
more than 10 years) and
a better estimate is to note that the handbook entry says that Total time commitment (should average) 170 hours. If we spread this over 14 weeks (12 class
and 2 weeks exam study), we get 12.14 hours per week, and then if we
subtract the number of class and tutorial hours per week (4) then we get
8.14 hours per week, and if we spread 12 weeks of 8 hours over 3 assignments
then we get
4*8.14=32.6, as the average number of hours of study and work
to complete each assignment.
If you are spending more time than this, then
please contact us
immediately
so that we can help you to make your study more efficient
and effective so that you
will be able to complete the necessary work for this class in
this expected time commitment timeframe.
If you are spending less time than this, then maybe you should think twice
about whether you will do well on the final exam.
The exam assesses whether you can write quality
solutions to questions in an exam setting. So the main goal of
the class is to learn to write quality, well presented solutions,
that communicate well and thoroughly to the reader. Whether or not
you get a correct answer has much less importance than
whether your exposition is of good quality.
To help learn this skill, we will provide three "proof writing" sessions
a week, in the same room as the lecture, in the hour before class.
These are not required, you decide whether you want to take advantage of
this resource or not. It is during these sessions that we will provide models
for writing solutions to exam questions.
Readings
The following notes written by Arun Ram and provide a good pcture of how
I think about and work with this subject.
Further references for useful comparison/contrast are:
Lectures and Tasks to keep up with the material 2022
The lectures will be recorded, but watching the recorded lectures is NOT
an efficient way to succeed in this course. The most efficient way to succeed
in this course is to make the ink flow out of the end of your pen in
the process of copying solutions and proofs. There are many solutions
and proofs provided as examples on this web page and on the web pages
for previous times that I have lectured this course.
 Lecture 1, 25 July 2022: Proof machine
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assignment 1 done. It is due on 11 August, which
is just over 2 weeks after the first lecture. Question 1 is a review question
from Linear algebra. Carefully, read the notes on
proof machine which are at this link
Proof Machine Notes
and look at some examples from the notes at this link
Inner products and Orthogonality notes
so that you can write a quality, thorough solution.
Do question 1 before Wednesday.
 Lecture 2, 27 July 2022: Function spaces
(i.e. infinite dimensional vector spaces)
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assigment 1 done. Since you finished question 1
already, do Question 2. Question 2 is a review question about convergence of
series from Real Analysis. Look at some examples from the notes at this
link Sequences and Series notes
so that you can write a quality, thorough solution.
Do question 2 before Friday.
 Lecture 3, 29 July 2022: Bounded linear operators
(i.e. linear transformations)
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assigment 1 done. Since you finished questions
1 and 2 already, do Question 3. Question 3 is a computation question that
helps to understand norms of operators.
Look at some examples from the notes at this
link Function spaces Notes
so that you can write a quality, thorough solution.
Do question 3 before Monday.
 Lecture 4, 1 August 2022: ℝ is the completion of ℚ
(i.e. What are the real numbers?)
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assigment 1 done. Since you finished questions
1 and 2 and 3 already, do Question 4 part A. Question 4A requires only basic
proof machine and a couple examples of series that are covered in Calculus 2.
To review the definition of a subset look at
Sets,
functions and relations Notes
and for the notion of what sup is look at
Orders and fields Notes.
Do question 4A before Wednesday.
 Lecture 5, 3 August 2022: Hilbert spaces
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assigment 1 done. Question 4B requires only basic
proof machine and the definitions of limits that are covered in Calculus 2 and
Real analysis. For a review of limits look at the notes at the
Limits Notes.
Do question 4B before Friday.
 Lecture 6, 5 August 2022: Completions
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assigment 1 done. Question 4F uses exactly
the same technique that was used to prove that ℝ is a completion
of ℚ in today's lecture.
For a review of Cauchy sequences and completion look at
Tutorial Sheet 1.
Do question 4F before Monday.
 Lecture 7, 8 August 2022: Hilbert space duals
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assigment 1 done. Question 4D uses exactly
the same technique that was used to prove that separable Hilbert space
is isomorphic to its dual in today's lecture.
For a review of this proof see
Tutorial Sheet 2
Do question 4D before Wednesday.
 Lecture 8, 10 August 2022: Orthogonality and projections I
Handwritten lecture notes
Student TODO list:
 The first goal is to get Assigment 1 done. Having done all the other
parts of Question 4, Question 4E is quite easy.
Finish up the assignment and submit it before 4pm on Thursday.
 Lecture 9, 12 August 2022: Orthogonality and projections II
Handwritten lecture notes
Student TODO list:
 Take a break over the weekend.
 Lecture 10, 15 August 2022: Eigenvalues and eigenvectors: compact operators
Handwritten lecture notes
Student TODO list:
 Lecture 11, 17 August 2022: Eigenvalues and eigenvectors: Self adjoint operators
Handwritten lecture notes
Student TODO list:
 We are now exploring the infinite dimensional versions of self adjoint opertors and the spectral theorem. Be sure to carefully review, and recopy by hand
the definitions and the proofs so that they are clear in your head,
Section 3.11 and 3.12 and Theorems 3.13 and 3.14 in the Linear algebra
review notes:
Inner Products and Orthogonality.
 Lecture 12, 19 August 2022: The spectral theorem for compact self adjoint operators
Handwritten lecture notes
Student TODO list:

I hope everyone had a good weekend and caught up on all your other classes.
 Lecture 13, 22 August 2022: A final lecture on spectral theorem and norms of operators
Handwritten lecture notes
Student TODO list:

As we finish up our section on Linear algebra, copy out the problems from
the sample exams that focus on linear algebra and do each of
them a few times as if you were training for an exam on this material.
 Lecture 14, 24 August 2022: Limits and neighborhoods
Handwritten lecture notes
Student TODO list:

Carefully read Limits and Topologies.
Do this by FIRST reading the references,
then reading SECTION FOUR (i.e. Section 4), then reading section 3, then reading section 2 and
then reading section 1.
 Lecture 15, 26 August 2022: Open, closed and neither  interior points and close points
Handwritten lecture notes
Student TODO list:

Now it is time to focus on Assignment 2. Read question 1 and remind yourself
what the precise definition of an infinite sum is.
Ask yourself where you've seen sums that look like this before.
Then put off question 1 and do questions 2 and 3, which are essentially
Calculus 2 questions. Finish Questions 2 and 3 by Monday.
 Lecture 16, 29 August 2022: Close points, cluster points and limit points
Handwritten lecture notes
Student TODO list:

The priority now is to finish Assignment 2.
Do the algebra parts of questions 4 and 5 (namely,
both part (c)s of question 4 and parts (b) and (c) of Question 5.
Get these done by Wednesday.
 Lecture 17, 31 August 2022:
Hausdorff spaces: Uniqueness of limits
Handwritten lecture notes
Student TODO list:

The priority now is to finish Assignment 2.
Do the graphing and integration parts of questions 4 (namely,
parts (a), (b), (d) and (e).
Get these done by Friday.
 Lecture 18, 2 September 2022: New topological spaces from old
Handwritten lecture notes
Student TODO list:

The priority now is to finish Assignment 2.
Do the rest of questions 4 and 5 (namely,
part (f) of question 4 and parts (a), (c) and (e) of question 5.
Get these done by Monday.
 Lecture 19, 5 September 2022: Compact spaces: Existence of limits
Handwritten lecture notes
Student TODO list:

The priority now is to finish Assignment 2.
Only one question left, which is question 1. It might help to review
the proof of Theorem 22.5 in
Tutorial sheet 2:
(and other results on this tutorial sheet) to knock this off.
Get this question done by Wednesday and submit the assignment in GRADESCOPE.
 Lecture 20, 7 September 2022: Compactness: Existence of cluster points
Handwritten lecture notes
Student TODO list:

The reading for this section is the notes on
Compact spaces.
Review these carefully. Show that z is not a cluster point
of a sequence (a_{1}, a_{2}, ...) if and only if there
exists B_{ε}(z) such that
B_{ε}(z) ∩ (a_{1}, a_{2}, ...) is finite.
 Lecture 21, 9 September 2022: Uniform convergence and pointwise convergence
Handwritten lecture notes
Student TODO list:

Spend time on your other classes this weekend.
Start making vocabulary flash cards for this course
to work through with friends, relative, acquaintances and random people
that you meet on public transport.
 Lecture 22, 12 September 2022: Uniform spaces and uniform continuity
Handwritten lecture notes
Student TODO list:

It is a good idea to get Assignment 3 done before the midsemester break.
Do parts (a) and (c) of Question 1 before Wednesday.
Spend any extra time you have working on your vocabulary flash cards.
 Lecture 23, 14 September 2022: Uniform spaces as a category
Handwritten lecture notes
Student TODO list:

It is a good idea to get Assignment 3 done before the midsemester break.
Do part (b) of Question 1 and part (a) of Question 2 before Friday.
Spend any extra time you have working on your vocabulary flash cards.
 Lecture 24, 16 September 2022:
Connected sets in ℝ
Handwritten lecture notes
Student TODO list:

It is a good idea to get Assignment 3 done before the midsemester break.
Do parts (b), (e) and (f) of Question 2 and part (a) of Question 3
and part (c) of Question 2 and part (b) of Question 3
before Monday.
Spend any extra time you have working on your vocabulary flash cards.
 Lecture 25, 19 September 2022:
Real numbers, padic numbers and formal power series
Handwritten lecture notes
Student TODO list:

It is a good idea to get Assignment 3 done before the midsemester break.
Do parts (a), (b) (c) of Question 4 and part (d) of Question 2 before Wednesday.
Spend any extra time you have working on your vocabulary flash cards.
 Lecture 26, 21 September 2022:
The compactopen topology, path spaces, fundamental groups, and connectedness
Handwritten lecture notes
Student TODO list:

It is a good idea to get Assignment 3 done before the midsemester break.
Do part (d) of Question 4 and start on Question 5 before Friday.
Spend any extra time you have working on your vocabulary flash cards.
 Lecture 27, 23 September 2022:
Grand Final Eve, NO CLASS
Student TODO list:

It is a good idea to get Assignment 3 done before the midsemester break.
Finish Question 5 before the game.
Spend any extra time you have working on and rewriting your vocabulary flash cards
over and over again.
 MIDSEMESTER BREAK
Student TODO list:

Spend any extra time you have working on and
rewriting your vocabulary flash cards
over and over again.
 Lecture 28, 3 October 2022:
The intermediate value theorem, the mean value theorem and Taylor's theorem
Handwritten lecture notes
Student TODO list:

Now it is time to intensively start preparing for the Exam.
Make a sample exam just using Tutorial sheets 1 and 2 and 3.
Make questions that look like the questions on the sample exams.
Do this by Wednesday.
 Lecture 29, 5 October 2022:
Properties of the real numbers
Handwritten lecture notes
Student TODO list:

Now it is time to intensively start preparing for the Exam.
Trade sample exams (from Tutorial sheets 1, 2 and 3)
that you made with a friend, and do each other's
exam and then mark your friend's work on your exam and discuss with
them how you can optimize your marks next time.
Do this by Friday.
 Lecture 30, 7 October 2022:
Limits via filters
Handwritten lecture notes
Student TODO list:

Do sample exam 4 over the weekend. Give yourself a 4 hour block
and allow yourself notes, but don't allow yourself to go overtime.
Mark your own work after you are done.
 Lecture 31, 10 October 2022:
Revisiting dual spaces
Handwritten lecture notes
Student TODO list:

Make a sample exam just using Tutorial sheets 4 and 5 and 6.
Make questions that look like the questions on the sample exams.
Do this by Wednesday.
 Lecture 32, 12 October 2022:
Compactness: training exercises
Handwritten lecture notes
Student TODO list:

Now it is time to intensively start preparing for the Exam.
Trade sample exams (from Tutorials 4, 5 and 6) that you made with a friend,
and do each other's exam and then mark your friend's work on your exam
and discuss with them how you can optimize your marks next time.
Do this by Friday.
 Lecture 33, 14 October 2022:
Generating topologies
Handwritten lecture notes
Student TODO list:

Do sample exam 4 over the weekend, the same one that you did last weekend.
This time only give yourself a 3 hour block
and do not allow yourself notes.
Mark your own work after you are done.
 Lecture 34, 17 October 2022:
Subspaces of ℝ^{∞}
Handwritten lecture notes
 Lecture 35, 19 October 2022:
Measures and Integration
Handwritten lecture notes
 Lecture 36, 21 October 2022:
Dual spaces revisited
Handwritten lecture notes
Typewritten lecture notes
Thoughts and advice
 The expectation is that the final exam will be
3 hours, IN PERSON, no additional notes allowed. If this changes
we will make an announcement of the change. Unless you can find some
announcement changing this plan on this webpage and on Canvas,
prepare for the case that no additional notes will be allowed on the exam.
 Just as a carpenter without a hammer is mostly useless, a mathematician
without a good place to work is mostly useless.
 Tips for time management:
 It is much easier (and safer) to run 45 min per day to attain 6 hours in a week and 24 hours in 4 weeks, than to run for 24 hours solid every fourth week on Sunday.
 To actually run 45 min, it takes me at least 15 min to psyche myself up and convince myself that it is actually not raining and so therefore I should go running, and after a 45 min run I always walk for 5 min and I always go home and have a glass of milk and tell my wife (at length) how cool I am for running 45 min per day. All in all, I waste a good 40 min when I go running for 45 min. If I were more efficient (and every so often, but rarely, I am) then it would only takes me 50 min.
 Measurement of time is a tricky thing and requires real discipline. Teaching and research faculty at University of Melbourne recently had to complete a survey on distribution of their time on the various activities of the job: Do I count the 6 times I had to go check my email and the weather and my iPhone in the time that I spend preparing my Metric and Hilbert Spaces lecture?
 Tips for exam preparation:
 The time that a 100m olympic runner (who wins a medal) is actually competing at the olympics is say (5 heats, 7sec each) 40 seconds. Successful performance during these 40 sec is impossible without adequate preparation.
 The time that a Metric and Hilbert spaces student spends on the final exam is 3 hours. Sucessful performance during these 3 hours is .....
 Prof. Ram reads email but generally does not respond by email.
Usually these are collated and reponses to email queries are provided
in the first few minutes of lectures. That way all students can benefit
from the answer to the query.
Feedback
If you don't communicate with us, then we can't help you. We'll do our best,
but it is much easier if you take some responsibility too by asking questions
and communicating in class and in tutorial.
Problem sheets
As explained in the writing
Teaching Math in The Next Life,
almost certainly, I will build the exam
by choosing the questions randomly from my problem lists for the course.
These problem lists for this course are the following.
Subject Overview
The handbook entry for this course is at
https://handbook.unimelb.edu.au/2022/subjects/mast30026. The subject overview that one finds there:
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.
Main Topics
 (1) Infinite dimensional vector spaces
 (2) Norms and distances
 (3) Inner products and orthogonality
 (4) Bounded and compact linear operators
 (5) The spectral theorem
 (6) Metric spaces, uniform spaces and Topological spaces
 (7) Limits, closure, continuity
 (8) Hausdorff and compact spaces
 (9) Cauchy sequences and completions
 (10) Connectedness and compactness for subsets of ℝ^{n}
 (11) Product topologies, quotient topologies and
function space topologies
 (12) Connections to probability and physics