Explaining mathematics

Latest posts on my Blogger blog 29/4/25

Joel — Tue, 29 Apr 2025 21:42:42 +0000

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.

Date	Title and link	Description
18/4/25	Some musings on Cauchy-Schwarz, Part Vb	A reminder (with more details) of why Cauchy-Schwarz for complex inner product spaces follows quickly from the result for real inner product spaces
18/4/25	Some musings on Cauchy-Schwarz, Part Va	The Cauchy-Schwarz inequality for general real inner product spaces follows immediately from the “easy” versions for real inner product spaces of dimension less than or equal to 2
18/4/25	Some musings on Cauchy-Schwarz, Part IVc	The case of two-dimensional real inner product spaces and how these are all essentially the same as with the usual dot product (and also when regarded as a two-dimensional real inner product space)
17/4/25	Some musings on Cauchy-Schwarz, Part IVb	Introductory discussion of the case of two-dimensional real inner product spaces, and a summary of what we have already said about them. Also some discussion of why I think it might possibly be worth spending all this time on such a well-known inequality, when there are short and elegant proofs available of the general case.
16/4/25	Some musings on Cauchy-Schwarz, Part IVa	The trivial case case of one-dimensional inner product spaces over
15/4/25	Some musings on Cauchy-Schwarz: Part III	The trivial case of one-dimensional inner product spaces over
8/4/25	Some musings on Cauchy-Schwarz: Part II	Connections between real and complex inner product spaces, and especially between and
8/4/25	Some musings on Cauchy-Schwarz: Part I	An introduction to a short series of posts looking at this well-known inequality: connections between the real and complex versions; Cauchy-Schwarz in dimensions less than or equal to 2.
26/2/24	Measures, outer measures and measurable sets I	A few initial thoughts about the relationships between measures and outer measures

Latest posts on my Blogger blog 5/2/25

Joel — Wed, 05 Feb 2025 18:55:29 +0000

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.

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

Date	Title and link	Description
5/2/25	The Extreme Value Theorem: a bootstrap approach	An approach to the Extreme Value Theorem where you can deduce all of the rest of the theorem from the first part where you show that every continuous function on a closed and bounded interval must be bounded above.
2/2/25	Set of real numbers, bounds and sequences II	In this part we (try to) tie together everything we know about infimum, supremum, epsilons and sequences
1/2/25	Set of real numbers, bounds and sequences I	Discussion of lower and upper bounds for sets, including the definitions of infimum and supremum
29/1/25	Injective continuous real-valued functions on an interval must be monotone	Some discussion of this very useful theorem of Real Analysis, and how to get around a minor obstacle in the usual proof by contradiction
11/12/24	A rule of thumb for when to use the Ratio Test	A discussion of when it might be a good idea to try the Ratio Test, and when you should definitely not try the Ratio Test (for sequences and/or series)

A rule of thumb for when to use the Ratio Test

Joel — Thu, 12 Dec 2024 12:12:15 +0000

My latest post on my Blogger blog is about when you should use the Ratio Test (for sequences and/or series) and, perhaps more importantly, when you should definitely NOT use the Ratio Test. I describe this briefly as a rule of thumb for when to use the Ratio Test

Latest posts on my Blogger blog 6/4/24

Joel — Sun, 07 Apr 2024 13:11:45 +0000

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.

Date	Title and link	Description
7/4/24	An introduction to the Hahn-Banach extension theorem: Part VI	Proof of the real Hahn-Banach theorem using transfinite induction (some details left to the reader).
6/4/24	An introduction to the Hahn-Banach extension theorem: Part V	Proof of the real Hahn-Banach theorem using Zorn’s Lemma
2/4/24	An introduction to the Hahn-Banach extension theorem: Part IV	How every complex normed space is also a real normed space, and how the complex and real dual spaces are related. Deduction of the complex Hahn-Banach theorem from the real Hahn-Banach theorem.
1/4/24	An introduction to the Hahn-Banach extension theorem: Part III	The Hahn-Banach Extension Theorem for linear functionals on separable, real normed spaces, proved using the key lemma and (at the end) extension to the closure.
31/3/24	An introduction to the Hahn-Banach extension theorem: Part II	– A look at the operator norm for bounded linear functionals on real normed spaces. – The key lemma used to extend bounded linear functionals on real normed spaces by one dimension without increasing the operator norm.
31/3/24	An introduction to the Hahn-Banach extension theorem: Part I	The statement of the Hahn-Banach extension theorem for bounded linear functionals on normed spaces, along with some initial comments about how we can approach the proof.

Latest posts on my Blogger blog 16/3/24

Joel — Sat, 16 Mar 2024 17:27:41 +0000

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
16/3/24	An introduction to the Weierstrass M-test: Part IV	Differentiation of power series term by term, based on the Weierstrass M-test.
14/3/24	An introduction to the Weierstrass M-test: Part III	A look at differentiation of series of functions term by term, based on the Weierstrass M-test.
11/3/24	A Cauchy-L’Hôpital-Taylor hybrid theorem	A two-function version of Taylor’s Theorem, from which you can deduce the usual version of Taylor’s Theorem with (Lagrange) remainder.
18/2/24	An introduction to the Weierstrass M-test: Part II	Proof of the Weierstrass M-test for series of real-valued functions defined on an interval: convergence and continuity of the resulting function under suitable conditions.
17/2/24	An introduction to the Weierstrass M-test: Part I	A preliminary look at series of real-valued functions defined on an interval, hopefully accessible to first-year mathematics undergraduate students.

Optional additional resources for first-year analysis

Joel — Thu, 14 Mar 2024 12:52:23 +0000

I’ve recently finished teaching 5 weeks of Analysis as part of our new first-year Core Mathematics module. I’ve made some additional resources available to them on a Moodle page. I expect to add further resources in due course! But here is the current version of that page.

Optional additional resources

Dr Feinstein’s blogs

I have two blogs about explaining mathematics. My WordPress blog at https://explainingmaths.wordpress.com/ has been running for a long time, but the version of LaTeX available there does not look as good as the more modern MathJax LaTeX available elsewhere these days. So I now also have a Blogger blog at https://explaining-maths.blogspot.com/

My two blogs do link to each other. Currently I write most of the mathematical posts on https://explaining-maths.blogspot.com/ and then post a link to those posts on https://explainingmaths.wordpress.com/

Taylor’s Theorem

This Blog post includes a two-function version of Taylor’s Theorem (similar to the Cauchy Mean Value Theorem, CMVT). This is something I thought about while teaching the material on differentiation in Analysis. It gives an alternative route in to proving Taylor’s Theorem via what I called a Hybrid Lemma, which resembles a combination of CMVT and L’Hopital’s rule. This two-function Taylor Theorem doesn’t appear to be widely known, and I suspect its main application is to prove the usual version of Taylor’s Theorem with (Lagrange form) remainder.

Mathematical Analysis

Another old module of mine, G12MAN Mathematical Analysis, is available in full for viewing or downloading from the University of Nottingham’s Open Educational Repository UNoW (http://unow.nottingham.ac.uk/). There is also a YouTube PlayList at https://www.youtube.com/playlist?list=PL58984C080F2B0575
Warning! In an attempt to help students understand some of the concepts better, I introduced some non-standard definitions: absorption of a sequence by a set; and the set of non-interior points in a set \(E\), denoted by \(\textrm{nint}\,E\,.\) Although potentially helpful for at least some students, these concepts have not caught on widely (yet!). So you should not use the terms absorption or \(\textrm{nint}\) in your work unless you include an explanation of what they mean. Your lecturers are unlikely to have heard of these terms!

Although much of the material concerns the topology of finite-dimensional Euclidean space, there is also some overlap with the analysis you are learning now. In particular, the chapters there on differentiation and on integration might be interesting (although the material on the Riemann integral is very compressed, and most of the results there are not proven). There is also a chapter on sequences of functions, where you can find out more about pointwise convergence and uniform convergence.

Introduction to the Weierstrass M-test

I am writing a series of posts on my Blog on the Weierstrass M-test.

An introduction to the Weierstrass M-test: Part I
A preliminary look at series of real-valued functions defined on an interval, hopefully accessible to first-year mathematics undergraduate students.
An introduction to the Weierstrass M-test: Part II
Proof of the Weierstrass M-test for series of real-valued functions defined on an interval: convergence and continuity of the resulting function under suitable conditions.

Core Topics in University Mathematics

Oldies but goldies! This playlist on YouTube includes 20 videos on topics that students often find challenging in first year. You’ll find some videos by me on Think of a function and on Sequences and their properties.

Foundations of Pure Mathematics

Here you can find some links to full sets of recordings of classes from various editions of a previous module of mine, Foundations of Pure Mathematics (FPM), which used to be the introductory module on pure mathematics in autumn semester of first year here. In particular, the recordings from the 2014 edition of FPM are available on YouTube Obviously things have changed a bit, but if you do want an alternative look at some of the topics, you are welcome to have a look!

Further Topics in Analysis optional additional resources

Joel — Thu, 14 Mar 2024 12:30:01 +0000

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!

Optional additional resources

My blogs
I have two blogs about explaining mathematics: one at https://explaining-maths.blogspot.com/ (which uses MathJax for the maths); and an older blog, with far more posts (but using a less modern version of LaTeX) at https://explainingmaths.wordpress.com/

My Open Access Resources on UNoW
The University of Nottingham has made some Open Access materials available via UNoW (http://unow.nottingham.ac.uk/). I was an active early adopter, and you can find quite a lot of my resources there. See https://rdmc.nottingham.ac.uk/handle/internal/79/most-popular/author# and https://rdmc.nottingham.ac.uk/discover?filtertype_1=author&filter_relational_operator_1=equals&filter_1=Feinstein,%20Joel&submit_apply_filter=&query=

Measure Theory
Videos and annotated slides from an old edition of Chapter 3 on Measure Theory are available via https://explainingmaths.wordpress.com/measure-theory/ (see also https://explainingmaths.wordpress.com/2022/05/16/measure-theory-playlists-on-mediaspace-and-youtube/)
These videos are also available as a YouTube Playlst at https://www.youtube.com/playlist?list=PLpRE0Zu_k-Bz6I691nDbE0qLsCFKwx5ku

Measure and Integration (G13MIN)
I used to give a 15-credit module (approx. 30 hours of classes) on Measure and Integration. You can find two years worth of annotated slides and audio recordings (mp3) from 2006-7-8, along with lecture materials, question sheets and solutions, etc., at http://tinyurl.com/UoNG13MIN

Functional Analysis (G14FUN)
Another former module of mine, Functional Analysis (G14FUN) is available in multiple formats. This module has a very large overlap with the current module MATH4047 Further Topics in Analysis, but G14FUN had more material on Functional Analysis and less material on Measure Theory.

There are two versions of G14FUN available for viewing and/or download from UNoW (https://unow.nottingham.ac.uk/): materials and audio recordings from 2006-7-8 at https://rdmc.nottingham.ac.uk/handle/internal/257; materials and videos from 2010 at https://rdmc.nottingham.ac.uk/handle/internal/269

There is also a G14FUN Playlist on YouTube at https://www.youtube.com/playlist?list=PL554B877A872B4F94

You may even be able to watch the videos with both Chinese and English subtitles/captions at http://open.163.com/special/opencourse/fanhanfenxi.html, though this appears to be blocked on the UoN network. (It worked from my home network, and on my mobile network today at least, 22/2/24)

Latest posts on my Blogger blog 11/3/24

Joel — Mon, 11 Mar 2024 14:25:13 +0000

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
11/3/24	A Cauchy-L’Hôpital-Taylor hybrid theorem	A two-function version of Taylor’s Theorem, from which you can deduce the usual version of Taylor’s Theorem with (Lagrange) remainder.
18/2/24	An introduction to the Weierstrass M-test: Part II	Proof of the Weierstrass M-test for series of real-valued functions defined on an interval: convergence and continuity of the resulting function under suitable conditions.
17/2/24	An introduction to the Weierstrass M-test: Part I	A preliminary look at series of real-valued functions defined on an interval, hopefully accessible to first-year mathematics undergraduate students.

8×8 Othello “solved”

Joel — Wed, 21 Feb 2024 18:41:45 +0000

This is amazing! I never thought I’d see this without some major breakthrough in computational power. It looks like, as long suspected, 8×8 Othello is a draw with perfect play.

https://arxiv.org/abs/2310.19387#:~:text=The%20challenge%20of%20solving%20Othello,players%20lead%20to%20a%20draw

See https://www.maths.nottingham.ac.uk/plp/pmzjff/Othello/6x6sol.html and https://www.maths.nottingham.ac.uk/plp/pmzjff/Othello/Amenor.html for what I established back in 1993 with my own (manually tuned, but somewhat amateurish) software on what was then our fastest departmental computer.

White to play and win 20-16!

Latest posts on my Blogger blog 18/2/24

Joel — Sun, 18 Feb 2024 13:29:14 +0000

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
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
26/12/23	Other approaches to connected components (from this year’s MATH4085 Metric and Topological Spaces)	An alternative approach to connected components, based on considering unions of just two connected sets with non-empty intersection, and then defining a suitable equivalence relation. (See also the earlier post on this.)
17/2/24	An introduction to the Weierstrass M-test: Part I	A preliminary look at series of real-valued functions defined on an interval, hopefully accessible to first-year mathematics undergraduate students.
18/2/24	An introduction to the Weierstrass M-test: Part II	Proof of the Weierstrass M-test for series of real-valued functions defined on an interval: convergence and continuity of the resulting function under suitable conditions.

Measure and Integration 2006-7-8 update

Joel — Wed, 14 Feb 2024 19:00:37 +0000

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.

Joel — Sun, 04 Feb 2024 18:03:48 +0000

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.

It’s going to take some time to sort all this out! But the screencasts (Foundations of Pure Mathematics, Mathematical Analysis, Functional Analysis, Measure Theory) are available on YouTube, and in some cases in a number of other places.

The audio files from Measure and Integration 2006-7-8 may be harder for me to track down, so I’ll have to ask for your patience on that. Hopefully they are not lost forever.

Note added 14/2/24

Thanks to the local team for helping me to track down the files, which were safely backed up!
I have (I think!) uploaded all of the Measure and Integration mp3 files to this blog, and have also set up a tinyurl to help me update the links. For example, the old url http://wirksworthii.nottingham.ac.uk/Podcasts/files/147680/MIN-1-2-08-c.mp3 (which no longer works for me) is now updated to be https://tinyurl.com/UoN-MIN-mp3/min-1-2-08-c.mp3 (which links to the mp3 file stored on this blog).

If you have any old links for my old Measure and Integration mp3 files, the main part of the URL should now be https://tinyurl.com/UoN-MIN-mp3/ followed by the file name … but also the filename is in lower case (so MIN becomes min, Lecture becomes lecture, and MP3 becomes mp3).

Nottingham Symphony Orchestra Concert 9th March 2024 (OT)

Joel — Sat, 20 Jan 2024 13:25:12 +0000

My wife Uta and I both play violin in the Nottingham Symphony Orchestra.

Our next concert, An evening of European Classics, will be on 9th March 2024, in the (Nottingham) Albert Hall.

Here is our flyer (as screenshots and, further down, as a PDF).

1739-nso-flyer-euro-classic-march-2024 Download

Piazza “less than” bug?

Joel — Sun, 14 Jan 2024 14:27:46 +0000

There appears to be a bug in (or a feature of) the Piazza mobile app (on android at least) that causes posts to be truncated at any strict less than sign inside maths.

It looks fine on a desktop web browser. (It is probably html-related.) But don’t try switching to the mark-up editor (even on the desktop web browser) as it literally deletes everything from the less than sign onwards!

Typing < inside maths in the Rich text editor on the desktop browser doesn’t help, because Piazza just converts it to the strict less than sign, which then breaks the app again.

Typing < in the Plain text editor on the desktop appears to be safe both for the desktop and for the mobile app.

This might be just a temporary bug/feature, but it may be just as well to be aware!

Note added 3 Feb 2024

I think a more efficient solution is to use the LaTeX code \lt instead of the less than sign.

Interestingly the greater than sign doesn’t appear to cause trouble. But < is probably interpreted as the start if of an html tag

I have notified Piazza of the issue, so hopefully they will fix this, or issue guidance.

Latest posts on my Blogger blog 26/12/23

Joel — Tue, 26 Dec 2023 16:57:22 +0000

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