2023 MAST30026 Metric and Hilbert Spaces

Many administrative details about the organisation of the subject can (as always) be found in the subject handbook entry.

Lecture notes

These are pretty much in final form now: current version (as of 28 October). The only updates that I might apply (if any) is corrections that may come up during consultation hours, and perhaps a few more exercises (I'll append these at the ends of chapters so that the numbering remains stable).

Tutorials

Tutorial classes start in week 2 of the semester. The tutorial sheets appear here at the start of the corresponding week, and solutions appear at the end of the week (at which point the exercises are also incorporated in the lecture notes).

Week 02: sheet and solutions
Week 03: sheet and solutions
Week 04: sheet and solutions
Week 05: sheet and solutions
Week 06: sheet and solutions
Week 07: sheet and solutions
Week 08: sheet and solutions
Week 09: sheet and solutions
Week 10: sheet and solutions
Week 11: sheet and solutions
Week 12: sheet and solutions

Assignments

The two assignments will be posted both here and on the subject's Canvas page. Your solutions should be submitted via Canvas and Gradescope.

The due dates for the assignments are: Friday 1 September at 8pm and Friday 13 October at 8pm.

Any requests for assignment special consideration should be sent to me, preferably not at the last minute.

Here is the first assignment, due Friday 1 September at 8pm. And here are the solutions for the first assignment.

Here is the second assignment, due Friday 13 October at 8pm. Note that the original version of the assignment had some errors that are corrected in the version linked here. And here are the solutions for the second assignment.

Exam preparation

The following topics are NOT examinable this year:

anything involving the $p$-adics;
anything requiring Lebesgue measure and Lebesgue integral (basically, the $L^p$ and $L^\infty$ spaces; note that the space of continuous functions on an interval and the $L^p$ and uniform norms on it are fair game though);
spectral theory (as we spent very little time on it).

As I mentioned in the lectures, the overwhelming majority (almost 90%) of exam marks involve questions that are taken from the subject material (in particular, that appear in the lecture notes file).

Please read the (very short) special exam coversheet to see what the exam conditions will be like.

I have been asked for recent past exams. Here are the ones from 2022 and 2021. I don't think you will find them very useful as preparation for this year's exam.

Ed discussion board

Please see the subject's Canvas page for access to the discussion board.

Lecture recordings

Please see the subject's Canvas page for access to the lecture recordings.

Other references: prerequisite knowledge

The main prerequisites for the subject are the University of Melbourne's MAST20022 Group Theory and Linear Algebra and MAST20026 Real Analysis (or some equivalent subject, see the handbook entry for details).

For those of you arriving with a different background, this means a solid understanding of linear algebra and previous exposure to abstract algebra concepts like groups, group actions, fields; also required is a firm grasp of analysis of functions on the real line.

There are many excellent abstract algebra and real analysis texts out there, so feel free to grab some to use as a reference while working on this subject.

Here are some suggestions:

Other references: metric and Hilbert spaces

There are also many excellent analysis texts out there covering various of the topics we are studying. I'll list here any that I refer to.

Analysis II by Tao

Note: Many of these references may be accessible via the library system either as electronic resources or physical tomes.