Course Information

Welcome to the course Banach spaces and locally convex spaces, presented through the NGA-Coursework platform. This is a more advanced course in functional analysis, that is aimed for MSc and PhD students who have completed a first course in functional analysis (all are welcome though).

Registration remains open. Please register on the NGA-Coursework website https://www.ngacoursework.org.za/ if you have not done so yet.

Lecturers

Course content

Below is roughly the content that we will discuss over 6 lectures.  

Course Content

We will roughly follow a selection of sections from Chapters III, IV, and V of the book: Conway, A course in functional analysis. 2nd Edition. Springer. 

It is likely that your home university library will have a hard copy that you can lend. (Shadow libraries like libgen do host electronic copies of many books, including Conway. If copyright law is something that you care to obey, you should be aware that shadow libraries do not adhere to copyright law. We neither encourage/discourage the use of shadow libraries).

Live Lecture Schedule

All activities will be online. Lectures will be delivered live. Unfortunately finding a timeslot that everyone can attend is usually impossible. Recordings of live lectures will be made available for participants who are unable to attend the live lectures.

We aim to have all live lectures on Thursdays at 10h00, but we may need to adjust the time if forced by loadshedding.

Lecture 1

Date : Thursday 2 March 2023

Time: 10h00 -- 12h00ish

Venue: Online

Lecture notes: [link]

Link:   https://youtube.com/live/0esaz9v46QY?feature=share

Lecture 2

Date : Thursday 9 March 2023

Time: 10h00 -- 12h00ish

Venue: Online

Lecture notes: [link]

Link:   https://youtube.com/live/ZGRcpMA-nGE?feature=share

Lecture 3

Date : Thursday 16 March 2023

Time: 10h00 -- 12h00ish

Venue: Online

Lecture notes: [link]

Link:   https://youtube.com/live/z_DhwJ-o1oM?feature=share

Lecture 4

Date : Thursday 23 March 2023

Time: 10h30 -- 12h30ish (We start 30 minutes later due to loadshedding)  10h00 -- 12h00ish

Venue: Online

Lecture notes: [link]

Link:   https://youtube.com/live/8b7R-d-OUEo?feature=share

Lecture 5

Date : Thursday 30 March 2023

Time: 10h00 -- 12h00ish

Venue: Online

Lecture notes: [link]

Link:   https://youtube.com/live/y78rsP59gbA?feature=share

Lecture 6

Date : Thursday 6 Apr 2023

Time: 10h30 -- 12h30ish (We start 30 minutes later due to loadshedding)  10h00 -- 12h00ish

Venue: Online

Lecture notes: [link]

Link:   https://youtube.com/live/6Syxi3H54yk?feature=share

Assessments

This course will be assessed through individual written assignments. Students are very strongly encouraged to discuss the assessments with fellow students and/or mentors. Writeups, however, should be completed and submitted individually. 

Handwritten submissions are accepted without penalty. As it is important for postgraduate students to become proficient in typesetting mathematical documents efficiently in LaTeX, students are encouraged to typeset their submissions in LaTeX, as this is excellent practice in learning how to use LaTeX. 

Assignment 01. Banach spaces part 1.

Assignment problems: [pdf]
Submission deadline: Fri 17 March 2023.

Some problems may be a bit tougher than others. Some hints are:
Q1: Use Zorn's Lemma.
Q3: Build a sequence of closed subsets that cover the entire space and use the Baire Category Theorem. 

Assignment 02. Banach spaces part 2.

Assignment problems: [pdf]
Submission deadline: Fri 24 March 2023.

Some problems may be a bit tougher than others. Some hints are:
Q5: Apply the the Hahn-Banach Theorem to construct a projection. Maybe see if you can get it to work for a 1 dimensional space, then a 2 dimensional space. 

Assignment 03. Locally convex spaces part 1.

Assignment problems: [pdf]
Submission deadline: Fri 7 April 2023.

Some problems may be a bit tougher than others. Some hints are interspersed in the pdf.

Assignment 04. Locally convex spaces part 2.

Assignment problems: [pdf]
Submission deadline: Fri 14 April 2023.

Some problems may be a bit tougher than others. Some hints are interspersed in the pdf.

We do monitor the Discord server on a daily basis, so if you are stuck, the please do not hesitate to ask for help on discord (or drop us an email). 

Assessment submission:

Please submit all assessments here : https://forms.gle/y17QQjtGrcA3i8ieA

Credit

Students hoping to gain official credit at their home university are required to discuss this possibility with their home university administration first and if there is agreement from the home university administration, then students should submit the required paperwork on the NGA-Coursework website (see the section: Application to receive official credit for Semester 1, 2023 course(s)). Final grades will be submitted to the home university administration, only if all written assessments were submitted before the stated deadlines. 

Discord server

https://discord.gg/g2s7N3E26R

Questions and live discussions

We will monitor the discord server and try to answer any questions.  If there is demand for having a live discussion session, then an ad hoc/or a regular online discussion session can be arranged. 

If possible, arranging a regular discussion session with fellow students and/or mentors at one's local university is very strongly encouraged.