Level 4 – Explaining mathematics

Latest posts on my Blogger blog 29/4/25

April 29, 2025September 21, 2025JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or click this search link

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

Currently I mostly write posts on my Blogger blog, and then link to them from this WordPress blog.

I’ll periodically post updates on this blog, with links and brief descriptions.

The latest series of posts is about the Cauchy-Schwarz inequality. Before that there was a post about measures, outer measures and measurable sets which I hope to follow up on once I have caught up on some urgent tasks.

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Latest posts on my Blogger blog 6/4/24

April 7, 2024November 14, 2024Joel1 Comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

My latest set of posts is a series of 6 posts (intended for mathematics undergraduates in their 3rd or 4th year) on the Hahn-Banach extension theorem for continuous linear functionals on normed spaces.

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Further Topics in Analysis optional additional resources

March 14, 2024November 14, 2024JoelLeave a comment

My Level 4 analysis module has been through various codes and titles. Currently it is MATH4047 Further Topics in Analysis (FTA for short).

I have made some optional additional resources available to them via a Moodle page.

Here is a copy of that page!

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Latest posts on my Blogger blog 18/2/24

February 18, 2024November 14, 2024JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Continue reading →

Measure and Integration 2006-7-8 update

February 14, 2024November 14, 2024JoelLeave a comment

Thanks to the local team at the University of Nottingham for helping me to track down my missing Measure and Integration mp3 files. I’m pleased to say that they were all safely backed up. I am now hosting the mp3 files on this blog.

Yesterday and today I’ve been working on the old Measure and Integration module page (from 2006-7-8) at http://tinyurl.com/UoNG13MIN, and I think that I have updated most of the links. But I might have missed a few, so please let me know if you spot any broken links!

Screencast and audio links from this blog.

February 4, 2024November 14, 2024JoelLeave a comment

Hi everyone,

An old server was recently decommissioned here at Nottingham (I was warned!)
I thought that I had safely transferred all of my screencasts and podcasts to somewhere sensible, and to some extent that is true. What I had forgotten was that this blog still has lots of links to the old server.

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Latest posts on my Blogger blog 26/12/23

December 26, 2023November 14, 2024JoelLeave a comment

For more of these summaries, you can search this blog for “Latest posts on my Blogger blog”, or directly see https://explainingmaths.wordpress.com/?s=%22Latest+posts+on+my+Blogger+blog%22

Here are some links to my most recent posts on my Blogger blog, where I am able to use MathJax.

I’ll periodically post updates on this blog, with links and brief descriptions.

Date	Title and link	Description
24/10/23	Converses and negations	Discussion of some specific examples to help understand the difference between converse and negation
24/10/23	Discussion of the proof that the uniform norm really is a norm	A detailed proof that the uniform really is a norm, including some comments and warnings on possible pitfalls along the way
23/12/23	Connectedness of unions	Discussion of a standard result concerning unions of connected sets, relevant to the theory of connected components of topological spaces
24/12/23	Totally bounded metric spaces	Discussion of the fact that a metric space X is totally bounded if and only if every sequence in X has a Cauchy subsequence. Also some initial discussion of uniformly separated sequences.
25/12/23	Sequences that have no Cauchy subsequences	Discussion of the fact that a sequence in a metric space has a Cauchy subsequence if and only if it has a subsequence which has no uniformly separated subsequence. Thus a sequence in a metric space has no Cauchy subsequence if and only if every subsequence has a uniformly separated subsequence.
25/12/23	The Christmas Equation	Taking a short break from total boundedness and Cauchy sequences, this is a repost of some funny maths I saw on QI a few years ago
25/12/23	Total boundedness for countable metric spaces	Some musings on the special case of countably infinite metric spaces. Not particularly conclusive! But there are various reformulations of the results above, and a note that imposing a well-order enables us to do without the usual sequence of choices. Also some discussion of uniformly separated subsets.
26/12/23	Conclusions on Totally Bounded metric spaces and Cauchy sequences	A summary of all the facts from the posts above, relating total boundness, uniformly separated sequences and subsets, and Cauchy subsequences

Discussion of the proof that the uniform norm really is a norm

October 25, 2023December 10, 2024JoelLeave a comment

In response to a request on Piazza, I gave a detailed proof that the uniform really is a norm, including some comments and warnings on possible pitfalls along the way. Here (essentially) is what I said.

This post is available from my Blogger blog at
https://explaining-maths.blogspot.com/2023/10/discussion-of-proof-that-uniform-norm.html
Alternatively, a PDF of that post is available for viewing or downloading below.

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Revisiting liminf and limsup

September 1, 2023December 10, 2024JoelLeave a comment

My recent series of posts about Fatou’s Lemma for sums was aimed primarily at mathematics students at 3rd-year undergraduate level or above. So I think I should write a post more suitable for first-year undergraduate students. So I’m going to have another look at the topic of $\liminf$ and $\limsup$ for bounded sequences of real numbers (though most of what I say can be generalised to sequences, or even nets, of extended real numbers).

This post is available from my Blogger blog at
https://explaining-maths.blogspot.com/2023/08/revisiting-liminf-and-limsup.html
Alternatively, a PDF of that post is available for viewing or downloading below.

Continue reading →