This blog may be moving back!

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In 2008 this blog moved to http://explainingmaths.wordpress.com
At the time WordPress offered better LaTeX support. But if MathJax works here, that may be an improvement on the standard LaTeX support on WordPress.com
Let's see how it goes!

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Some musings on Cauchy-Schwarz, Part Vb

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 Recall our claim from Part I Cauchy-Schwarz in complex inner product spaces can be deduced from Cauchy-Schwarz in real inner product spaces. This is essentially because every complex inner product space can also be regarded as a real inner product space with the same norm, but where the real inner product is the real part of the complex inner product. The last bit of the deduction uses a fairly standard "rotation trick": for any complex number \(z\), we have  \[|z|=\max\{\Re(\alpha z):\alpha \in \C, |\alpha|=1\,\}\,.\qquad(*)\] Let's check $(*)$ first. Let $z\in\C$, and set \[A=\{\Re(\alpha z):\alpha \in \C, |\alpha|=1\,\} \subseteq \R\,.\]  Clearly $A$ is non-empty.  Let $\alpha \in \C$ with $|\alpha|=1$. Then certainly we have \[\Re(\alpha z) \leq |\alpha z| = |z|\,.\] Thus we have \[\sup A \leq |z|\,.\] We now need to prove that equality holds, and that the supremum is actually a maximum. For this we just need to show that $|z| \in A$. This is trivial if $z=0$, b...
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A rule of thumb for when to use the Ratio Test

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The Ratio Test is a powerful test for both sequences and series of non-zero real numbers. It comes in various forms, but here is one commonly used version. Theorem (Ratio Test) Let $(x_n)_{n\in\N}$ be a sequence of non-zero real numbers. Suppose that \[\frac{|x_{n+1}|}{|x_n|}\to L \text{ as } n\to\infty\,,\] where $L$ is either a non-negative real number or $+\infty$. (a) If $L>1$, then $|x_n| \to +\infty$ as $n \to + \infty$, the sequence $(x_n)$ is divergent, and the series $\displaystyle \sum_{n=1}^\infty x_n$ diverges. (b) If $L \in [0,1)$, then the series $\displaystyle \sum_{n=1}^\infty x_n$ is absolutely convergent (and hence convergent), and the sequence $(x_n)$ converges to $0$. (c) If $L=1$, then the Ratio Test is inconclusive , and you need to use a different test . (The Ratio Test tells you nothing in this case about the convergence or otherwise of either the sequence or the series.) Notes Here (b) does not tell you anything about the value of the sum of the series ...
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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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