Explaining maths

In this post we establish (in Lemma 2 below) the "key step" needed for the Hahn-Banach theorem (as described in the previous post), where we extend our linear functional by one real dimension. First we note an elementary fact about the operator norm for continuous linear functionals on real normed spaces. Lemma 1 (Operator norm for real linear functionals) Let $(E,\jnorm)$ be a normed space over $\R$, and let $\phi \in E^*.$ Then $(1)\Jdisplay\|\phi\|_\op = \sup \{\phi(x): x \in E, \|x\| \leq 1\,\}\,.$ Comment This looks very like the definition of the operator norm! However the subtle difference is that we work with $\phi(x)$ instead of $|\phi(x)|$ here. Our actual definition of the operator norm is that \[\|\phi\|_\op = \sup \{|\phi(x)|: x \in E, \|x\| \leq 1\,\}\,.\] Note that this would make no sense, as it stands, for complex normed spaces (though we could work with the real part of $\phi(x)$ in that case, and we would again obtain a true result). Let us refer to the ...

In this series of posts (intended for mathematics undergraduate students in their 3rd or 4th year) I am going to focus on the Hahn-Banach extension theorem for bounded linear functionals on normed spaces. I may also say something about more general versions in terms of sublinear functions.  Let $\F$ be $\R$ or $\C$, and let $(E,\jnorm)$ be a normed space over $\F$. We denote by $E^*$ the set of continuous (bounded) linear functionals on $E$, that is, the set of continuous linear maps from $(E,\jnorm)$ to $(\F,|\cdot|).$ Since $(\F,|\cdot|)$ is complete, it follows that $E^*$ is always complete when given the operator norm  $\jnorm_\op$ defined, for $\phi\in E^*$, by \[\|\phi\|_\op = \sup\{|\phi(x)|:x \in E, \|x\|\leq 1\,\}\,.\] Now let $F$ be a linear subspace of $E,$ and give $F$ the norm obtained by restricting $\jnorm$ to $F.$ We still denote this restricted norm by  $\jnorm$. We now have another dual space $F^*,$ along with a restriction map $R:E^*\to F^*$ defin...

I think this will be the last post in this series. This time we will look at the special case of power series, using Theorem 5 from the previous part to help us to justify differentiation term by term. Notation:  Recall that we denote the set  \(\N \cup \{0\}\) of  non-negative  integers by \(\N_0.\) Recall the statement of our Theorem 5. Theorem 5 (Term by term differentiation of series) Let $I$ be a nondegenerate interval in $\R$, and let $M_n$ ($n \in \N_0$) be non-negative real numbers such that $\sum_{n=0}^\infty M_n < \infty\,.$ Let $f_n$ ($n \in \N_0$) be differentiable functions from $I$ to $\R,$ and suppose that, for all $n \in \N_0$ and all $x \in I,$ we have $|f_n'(x)|\leq M_n.$ Suppose further that, for  all $x \in I$, the series $\sum_{n=0}^\infty f_n(x)$ converges. Define $f:I \to \R$ by \[ f(x)=\sum_{n=0}^\infty f_n(x)\,.\] Then $f$ is differentiable on $I$, and, for each $x \in I$, \[f'(x)=\sum_{n=0}^\infty f_n'(x)\,.\] We are now re...

In this post, following on from the first two posts in the series, we look at differentiating suitable series of functions term by term. We make as much use as possible of the Weierstrass M-test (in the spirit of this series of posts), instead of discussing uniform convergence in general. Of course, uniform convergence is an important topic, and I'll come back to it at some point. First we recall a characterization of differentiability in terms of "chord functions". Note, for functions defined on intervals, when we discuss differentiability at endpoints (if any) of the interval, we use one-sided derivatives. Proposition 3 (Differentiability in terms of chord functions) Let $I$ be a non-degenerate interval in $\R$, let $f:I\to \R$, and let $x_0 \in I.$ Then the following statements are equivalent. (a) The function $f$ is differentiable at $x_0$. (b)  There is a function $g:I \to \R$ such that $g$ is continuous at $x_0$ and, for all $x \in I$, \[f(x) = f(x_0) + (x-x_0) g(x...

Last week I was teaching the standard theorems of real analysis (single variable): Rolle's Theorem, the Mean Value Theorem (MVT), Cauchy's Mean Value Theorem, L'Hôpital's rule, and Taylor's Theorem. As I was short of time, I didn't manage to give full proofs of everything live (but they have complete proofs in the printed notes). I did however have a think about the relationships between these results, and what I might say about them. Along the way, I noticed that there is a particular hybrid Cauchy-L'Hôpital-Taylor result which I don't remember seeing before (though no doubt it is well known). Let me start by recalling some of the standard statements. For the rest of this post , \(a\) and \(b\) are real numbers with \(a<b,\) and \(f\) and \(g\) are real-valued functions on \([a,b]\) which are both continuous on \([a,b]\) and differentiable on \((a,b).\) Rolle's Theorem Suppose that \(f(a)=f(b)\). Then there exists \(\xi \in (a,b)\) suc...