Selected posts from my WordPress blog

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Since I can now use MathJax here, but not on my WordPress blog, I am going to transfer some of my recent posts over from WordPress, using MathJax here. But first I have a lot of marking to do!

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Sums and integration against counting measure: Part I

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For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I have made a PDF of the main article available via my  WordPress blog  at  https://explainingmaths.files.wordpress.com/2023/08/sums-and-integration-against-counting-measure_-part-i.pdf Note added 30/8/23: I'm not quite sure when I started to say "integration against a measure". I'm not alone, but I think that "integration with respect to a measure" is a lot more common! However, I don't want to change the titles of my latest 10 blog posts, so I'll leave it as it is. I'll probably change back to "integration   with respect to a measure" in future though! It has been a while since I wrote a mathematical post! Here I thought I would say something about Measure and Integration which is relevant to my research. For these posts I as
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Discussion of the proof that the uniform norm really is a norm

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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. Let’s have a look at \(\|\cdot\|_\infty\) on the bounded, real-valued functions on \([0,1]\) (but you can also work with bounded functions on any non-empty set you like, and the same proof will work). We define \(\ell_\infty([0,1]) = \{f: f \textrm{ is a bounded, real-valued function on }[0,1]\,\}\,,\) and this is the set/space we will look at. So we set \(Y=\ell_\infty([0,1])\). Then you can check for yourselves that \(Y\) really is a vector space over \(\mathbb{R}\) (using the usual "pointwise" operations for functions). Note that the zero element in this vector space is the constant function \(0\) (from \([0,1]\) to \(\mathbb{R}\)), which I'll denote by \(\mathbf{0}\) here for clarity. (We also used \(Z\) for this function in one of the exercises.) I'll first give
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Revisiting liminf and limsup

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This post is also available in PDF form from my WordPress Blog at  https://explainingmaths.wordpress.com/2023/09/01/revisiting-liminf-and-limsup/ 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). I said quite a lot about this in a previous post on my WordPress blog in the post https://explainingmaths.wordpress.com/2009/01/10/an-application-of-absorption-to-teaching-lim-inf-and-lim-sup/ and there was a lot of discussion in the comments below the post. In particular, in those comments, I discussed the $\liminf$ and the $\limsup$ of some examples of sequences $(x_n)$, including the following exam
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