0$ we can find a $g\in C([-1,1])$ such that $\|g\|_{2}\geq K\|g\|_{1}$. \end{question} \vspace{1\baselineskip} The remaining question is included for general interest and not for relevance to the syllabus or to passing Tripos exams. \vspace{1\baselineskip} \begin{question} We use the notation of Questions~\ref{3.10} and~\ref{3.11}. (i) Show that if $\alpha\in\mathcal{E}$ we can find $e^{\alpha}\in\mathcal{E}$ such that \[\left\|\sum_{r=0}^{n}\frac{\alpha^{r}}{r!} -e^{\alpha}\right\|\rightarrow 0\] as $n\rightarrow\infty$. (ii) Show carefully that if $\alpha$ and $\beta$ commute \[e^{\alpha}e^{\beta}=e^{\alpha+\beta}.\] (iii) Show that if $\alpha$ and $\beta$ are general (not necessarily commuting) elements of ${\mathcal E}$ then \[\left\|h^{-2}(e^{h\alpha}e^{h\beta}-e^{h\beta}e^{h\alpha}) -(\alpha\beta-\beta\alpha)\right\|\rightarrow 0\] as the real number $h\rightarrow 0$. Conclude that, in general, $e^{\alpha}e^{\beta}$ and $e^{\alpha+\beta}$ need not be equal. \end{question} \newpage \section{Fourth Sheet of Exercises} I have tried to produce 12 rather routine questions for each sheet. The remaining questions are for interest only. Ambitious students and their supervisors should look at Tripos questions to supplement this meagre fare. \vspace{1\baselineskip} \begin{question}\label{Odd railway} Consider the two railway metrics $d_{1}$ and $d_{2}$ on ${\mathbb R}^{2}$ given in Examples~\ref{Railway 1} and~\ref{Railway 2}. For each of the two metrics give a reasonably simple description of the open balls of radius $r$ and centre ${\mathbf x}$ when ${\mathbf x}={\mathbf 0}$ and when ${\mathbf x}\neq{\mathbf 0}$ and (a)~$r\leq\|{\mathbf x}\|$, (b)~$r<\|{\mathbf x}\|\leq 2r$, (c)~$2r<\|{\mathbf x}\|$. Explain the statement `when finding open sets only balls of small radius are important so cases (b) and (c) are irrelevant'. Give the simplest description you can find of open sets in the two metrics. \end{question} \begin{question} Let $f:{\mathbb R}^{2}\rightarrow{\mathbb R}$ be differentiable and let $g(x)=f(x,c-x)$ where $c$ constant. Show that $g:{\mathbb R}\rightarrow{\mathbb R}$ is differentiable and find its derivative (i) directly from the definition of differentiability \noindent and also (ii) by using the chain rule. \noindent Deduce that if $f_{,1}=f_{,2}$ throughout ${\mathbb R}$ then $f(x,y)=h(x+y)$ for some differentiable function $h$. \end{question} \begin{question} [Traditional] Consider the functions $f_{n}:[0,1]\rightarrow{\mathbb R}$ defined by $f_{n}(x)=n^{p}x\exp(-n^{q}x)$ where $p,q>0$. (i) Show that $f_{n}$ converges pointwise on $[0,1]$. (ii) Show that if $p0$ such that $\|\mathbf{x}\|\geq k$ for all $\mathbf{x}$ with $\|\mathbf{x}\|_{2}=1$. Conclude that \[k\|\mathbf{x}\|_{2}\leq \|\mathbf{x}\|\leq K\|\mathbf{x}\|_{2}\] for all $\|\mathbf{x}\|\in {\mathbb R}^{m}$. \noindent [Thus all norms on a finite dimensional space are essentially the same.] (ii) Consider the real vector space $\mathbb{R}^{\mathbb{N}}$ of all real sequences $\mathbf{x}=(x_{1},x_{2},\dots)$ with the usual vector addition and multiplication by scalars. Let $V$ be the set of all sequences $\mathbf{x}=(x_{1},x_{2},\dots)$ such that $x_{j}\neq 0$ for only finitely many $j$. Show that $V$ is a subspace of $\mathbb{R}^{\mathbb{N}}$ and so a vector space in its own right. By considering norms of the form $\|\mathbf{x}\|=\max_{j}\kappa_{j}|x_{j}|$, or otherwise, find two norms $\|\ \|_{A}$ and $\|\ \|_{B}$ such that \[\sup_{\|\mathbf{x}\|_{A}=1}\|\mathbf{x}\|_{B} =\sup_{\|\mathbf{x}\|_{B}=1}\|\mathbf{x}\|_{A}=\infty.\] \end{question} \begin{question} (i) Suppose that $f:{\mathbb R}\rightarrow{\mathbb R}$ is a twice differentiable function such that \[\left|\frac{f(x)f''(x)}{f'(x)^{2}}\right| \leq\lambda\] for all $x$ and some $|\lambda|<1$ Show that the mapping \[Tx=x-\frac{f(x)}{f'(x)}\] is a contraction mapping and deduce that $f$ has a unique root $y$. (ii) Suppose that $F:{\mathbb R}\rightarrow{\mathbb R}$ is a twice differentiable function such that \[\left|\frac{F(x)F''(x)}{F'(x)^{2}}\right| \leq\lambda\] for all $|x|\leq a$ and some $|\lambda|<1$ and that $F(0)=0$. Consider the mapping \[Tx=x-\frac{F(x)}{F'(x)}.\] Show that $T^{n}x\rightarrow 0$. Suppose that \[\frac{\sup_{|t|\leq a}|f'(t)|\sup_{|t|\leq a}|f''(t)|} {\inf_{|t|\leq a}|f'(t)|^{2}}=M.\] By using the mean value theorem twice, show that if $|x|\leq a$ then \[|Tx|\leq Mx^{2}.\] (iii) If you know what the Newton--Raphson method is, comment on the relevance of the results of (i) and (ii) to that method. \end{question} \begin{question} We work in ${\mathbb R}^{m}$ with the usual `Euclidean norm'. The following line of thought is extremely important in later work. Suppose we want the solution ${\mathbf x}_{0}$ of \[{\mathbf x}+{\boldsymbol\epsilon}({\mathbf x})= {\mathbf y}\] where ${\boldsymbol\epsilon}({\mathbf x})$ is `a small error term' or `of first order compared to ${\mathbf x}$. The following method of solution is traditional (for the excellent reason that it usually works like a Spanish charm). Start by ignoring the small ${\boldsymbol\epsilon}$ term and guess ${\mathbf x}_{0}\approx {\mathbf x}_{1}={\mathbf y}$. Since this initial guess is good we estimate the small ${\boldsymbol\epsilon}$ term as ${\boldsymbol\epsilon}({\mathbf x}_{1})$. Feeding this estimate back into the equation we now guess ${\mathbf x}_{0}\approx {\mathbf x}_{2}={\mathbf y}-{\boldsymbol\epsilon}({\mathbf x}_{1})$. Repeating this process gives us the rule for successive approximations \[\mathbf{x}_{n+1}=\mathbf{y}-{\boldsymbol\epsilon}(\mathbf{x}_{n}).\] We now try to justify this in certain cases. (i) Suppose that ${\boldsymbol\epsilon}({\mathbf 0})={\mathbf 0}$ and \[\|{\boldsymbol\epsilon}({\mathbf a})- {\boldsymbol\epsilon}({\mathbf b})\| <\|{\mathbf a}-{\mathbf b}\|/2\] for all $\|\mathbf{a}\|,\|\mathbf{b}\|\leq\delta$ where $\delta>0$. (Note that these are rather stronger conditions than might have been expected. However, this is not to say that the general idea might not work under weaker conditions.) We shall show that the method works for $\|{\mathbf y}\|\leq\delta/2$. Consider the closed ball $B=\{{\mathbf x}:\|{\mathbf x}\|\leq \delta\}$. Show that the equation \[T({\mathbf x})=\mathbf{y}-{\boldsymbol\epsilon}({\mathbf x})\] defines a map $T:B\rightarrow B$. Now use the contraction mapping theorem to show that \[{\mathbf x}+{\boldsymbol\epsilon}({\mathbf x})= {\mathbf y}\] has a unique solution ${\mathbf x}(\mathbf{y})$ with $\|{\mathbf x}(\mathbf{y})\|\leq\delta$ and that $T^{n}({\mathbf y})\rightarrow {\mathbf x}(\mathbf{y})$ as $n\rightarrow\infty$. (ii) Suppose that $f:{\mathbb R}^{m}\rightarrow{\mathbb R}^{m}$ is a differentiable function with derivative continuous at ${\mathbf 0}$, $f({\mathbf 0})={\mathbf 0}$ and $Df({\mathbf 0})=I$. Show that we can find a $\delta>0$ such that whenever $\|{\mathbf y}\|\leq\delta/2$ the equation \[f({\mathbf x})={\mathbf y}\] has a unique solution ${\mathbf x}(\mathbf{y})$ with $\|{\mathbf x}(\mathbf{y})\|\leq\delta$. (iii) Suppose that $f:{\mathbb R}^{m}\rightarrow{\mathbb R}^{m}$ is a differentiable function with derivative continuous at ${\mathbf 0}$, $f({\mathbf 0})={\mathbf 0}$ and $Df({\mathbf 0})$ invertible. Show that we can find a $\delta_{1},\delta_{2}>0$ such that whenever $\|{\mathbf y}\|\leq\delta_{1}$ the equation \[f({\mathbf x})={\mathbf y}\] has a unique solution ${\mathbf x}(\mathbf{y})$ with $\|{\mathbf x}(\mathbf{y})\|\leq\delta_{2}$. \end{question} \begin{question} (A method of Abel) (i) Suppose that $a_{j}$ and $b_{j}$ are sequences of complex numbers and that $S_{n}=\sum_{j=1}^{n}a_{j}$ for $n\geq 1$ and $S_{0}=0$. Show that, if $1\leq u\leq v$ then \[\sum_{j=u}^{v}a_{j}b_{j}=\sum_{j=u}^{v}S_{j}(b_{j}-b_{j+1}) -S_{u-1}b_{u}+S_{v}b_{v+1}.\] (This is known as partial summation, for obvious reasons.) (ii) Suppose now that, in addition, the $b_{j}$ form a decreasing sequence of positive terms and that $|S_{n}|\leq K$ for all $n$. Show that \[\left|\sum_{j=u}^{v}a_{j}b_{j}\right| \leq 2Kb_{u}.\] Deduce that if $b_{j}\rightarrow 0$ as $j\rightarrow\infty$ then $\sum_{j=1}^{\infty}a_{j}b_{j}$ converges. Deduce the alternating series test. (iii) If $b_{j}$ is a decreasing sequence of positive terms with $b_{j}\rightarrow 0$ as $j\rightarrow\infty$ show that $\sum_{j=1}^{\infty}b_{j}z^{j}$ converges uniformly in the region given by $|z|\leq 1$ and $|z-1|\geq \epsilon$ for all $\epsilon>0$. \end{question} \begin{question}\label{4.9} We work in ${\mathbb R}^{m}$ with the usual distance. (i) Show that if $A_{1}$, $A_{2}$, are non-empty, closed and bounded with $A_{1}\supseteq A_{2}\supseteq\dots$ then $\bigcap_{j=1}^{\infty}A_{j}$ is non empty. Is this result true if we merely assume $A_{j}$ closed and non-empty? Give reasons. (ii) If $A$ is non-empty, closed and bounded show that we can find ${\mathbf a}',{\mathbf b}'\in A$ such that \[\|{\mathbf a}'-{\mathbf b}'\|\geq\|{\mathbf a}-{\mathbf b}\|\] for all ${\mathbf a},{\mathbf b}\in A$. Is this result true if we merely assume $A$ bounded and non-empty? Give reasons. \end{question} \begin{question} We work in ${\mathbb R}^{m}$ with the usual distance. Let $E$ be a closed non-empty subset of ${\mathbb R}^{m}$ and let $T$ be a map $T:E\rightarrow E$. (i) Suppose $\|T({\mathbf a})-T({\mathbf b})\|<\|{\mathbf a}-{\mathbf b}\|$ for all ${\mathbf a},{\mathbf b}\in E$ with ${\mathbf a}\neq{\mathbf b}$. We saw in Example~\ref{no fixed} that $T$ need not have a fixed point. Show that, if $T$ has a fixed point, it is unique. (ii) Suppose $\|T({\mathbf a})-T({\mathbf b})\|>\|{\mathbf a}-{\mathbf b}\|$ for all ${\mathbf a},{\mathbf b}\in E$ with ${\mathbf a}\neq{\mathbf b}$. In Question~\ref{4.16} it is shown that $T$ need not have a fixed point. Show that, if $T$ has a fixed point, it is unique. (iii) Suppose $\|T({\mathbf a})-T({\mathbf b})\|=\|{\mathbf a}-{\mathbf b}\|$ for all ${\mathbf a},{\mathbf b}\in E$. Show that $T$ need not have a fixed point. and that, if $T$ has a fixed point, it need not be unique. (iv) Suppose now that $E$ is non-empty, closed and bounded and \[\|T({\mathbf a})-T({\mathbf b})\|<\|{\mathbf a}-{\mathbf b}\|\] for all ${\mathbf a},{\mathbf b}\in E$ with ${\mathbf a}\neq{\mathbf b}$. By considering $\inf_{{mathbf x}\in E}\|{\mathbf x}-T({mathbf x})\|$, or otherwise show that $T$ has a fixed point. \end{question} \begin{question} From time to time numerical analysts mention the \emph{spectral radius}. It forms no part of any of non-optional 1B courses but the reader may be interested to see what it is. (i) Give an example of a linear map $\beta:{\mathbb R}^{m}\rightarrow{\mathbb R}^{m}$ such that $\beta^{m-1}\neq {\mathbf 0}$ but $\beta^{m}={\mathbf 0}$. (ii) Let $\alpha:{\mathbb R}^{m}\rightarrow{\mathbb R}^{m}$ be linear. If $n=jk+r$ explain why \[\|\alpha^{jk+r}\|\leq \|\alpha^{j}\|^{k}\|\alpha\|^{r}.\] (iii) Continuing with the hypotheses of (ii), show that $\Delta=\inf_{n}\|\alpha^{n}\|^{1/n}$ is well defined and, by using the result of (ii), or otherwise, that \[\|\alpha^{n}\|^{1/n}\rightarrow \Delta.\] We call $\Delta$ the spectral radius of $\alpha$ and write $\rho(\alpha)=\Delta$. (iv) If $\alpha:{\mathbb R}^{m}\rightarrow{\mathbb R}^{m}$ is diagonalisable show that \[\rho(\alpha)=\max\{|\lambda|:\lambda\ \text{an eigenvalue of $\alpha$}\}.\] (v) Give an example of linear maps $\alpha,\beta:{\mathbb R}^{2}\rightarrow{\mathbb R}^{2}$ such that $\rho(\alpha)=\rho(\beta)=0$ but $\rho(\alpha+\beta)=1$. (vi) Recalling Question~\ref{3.11} of the third sheet, show that if $\rho(\iota-\gamma)<1$ then $\gamma$ is invertible. \end{question} \vspace{1\baselineskip} The remaining questions are included for general interest and not for relevance to the syllabus or to passing Tripos exams. \begin{question}\label{4.12} We work in ${\mathbb R}^{m}$, $\|{\mathbf x}-{\mathbf y}\|$ will represent the usual Euclidean distance between ${\mathbf x}$ and ${\mathbf y}$. (i) If $K$ is a closed non-empty bounded set and ${\mathbf x}$ is any point, show that there exists a point ${\mathbf k}'\in K$ such that \[\|{\mathbf x}-{\mathbf k}\|\geq \|{\mathbf x}-{\mathbf k}'\|\] for all ${\mathbf k}\in K$. Is ${\mathbf k}'$ necessarily unique? (ii) If $E$ is a non-empty closed set and ${\mathbf x}$ is any point, show that there exists a point ${\mathbf e}'\in E$ such that \[\|{\mathbf x}-{\mathbf e}\|\geq \|{\mathbf x}-{\mathbf e}'\|\] for all ${\mathbf e}\in E$. We write $d({\mathbf x},E)=\|{\mathbf x}-{\mathbf e}'\|.$ (iii) With the notation of (ii) show that \[d({\mathbf x},E)+\|{\mathbf x}-{\mathbf y}\| \geq d({\mathbf y},E).\] Show that $d(\ ,E):{\mathbb R}^{m}\rightarrow{\mathbb R}$ is continuous. (iv) If $E$ is closed and non-empty and $K$ closed, bounded and non-empty show that there exist ${\mathbf e}'\in E$ and ${\mathbf k}'\in K$ such that \[\|{\mathbf e}-{\mathbf k}\|\geq \|{\mathbf e}'-{\mathbf k}'\|.\] Would this result be true if we only assumed $E$ and $K$ closed and non-empty. \end{question} \begin{question}\label{4.13} (i) Show that if $f:{\mathbb R}\rightarrow{\mathbb C}$ is a continuous periodic function with period $2\pi$ such that $\hat{f}(n)=0$ for all $n$ then \[\int_{-\pi}^{\pi}(1-\epsilon_{1}+\epsilon_{2}\cos t)^{N}f(t)\,dt=0\] for all $N$. (ii) Show that, given $\delta>0$ we can find $\epsilon_{1},\ \epsilon_{2}>0$ and an $\eta>0$ such that \begin{align*} (1-\epsilon_{1}+\epsilon_{2}\cos t)^{N}\rightarrow\infty&\ \text{uniformly for $|t|\leq\eta$}\\ (1-\epsilon_{1}+\epsilon_{2}\cos t)^{N}\rightarrow 0&\ \text{uniformly for $\delta\leq |t|\leq\pi$} \end{align*} as $N\rightarrow\infty$. (iii) Show that if $f:{\mathbb R}\rightarrow{\mathbb C}$ is a continuous periodic function with period $2\pi$ such that $f(0)$ is real and $f(0)>0$ we can find $\epsilon_{1},\ \epsilon_{2}>0$ such that \[\Re\int_{-\pi}^{\pi}(1-\epsilon_{1}+\epsilon_{2}\cos t)^{N}f(t)\,dt \rightarrow\infty\] as $N\rightarrow\infty$. (iv) Show that if $f:{\mathbb R}\rightarrow{\mathbb C}$ is a continuous periodic function with period $2\pi$ such that $\hat{f}(n)=0$ for all $n$ then $f(t)=0$ for all $t$. (v) By using Question~\ref{4.3}, or otherwise, show that if if $F:{\mathbb R}\rightarrow{\mathbb C}$ is a continuous periodic function with period $2\pi$ such that $\sum |\hat{F}(n)|$ converges then \[F(t)=\sum_{n=-\infty}^{\infty}\hat{F}(n)\exp (int).\] \end{question} \begin{question} (i) Suppose that $g:{\mathbb R}\rightarrow{\mathbb C}$ is a continuous periodic function with period $2\pi$. Show that \[\frac{1}{2\pi}\int_{0}^{2\pi} \left|g(t)-\sum_{r=-N}^{N}\hat{g}(r)\exp(irt)\right|^{2} \,dt =\frac{1}{2\pi}\int_{0}^{2\pi}|g(t)|^{2}\,dt -\sum_{r=-N}^{N}|\hat{g}(r)|^{2}.\] Deduce that \[\sum_{r=-N}^{N}|\hat{g}(r)|^{2} \leq\frac{1}{2\pi}\int_{0}^{2\pi}|g(t)|^{2}\,dt.\] (This is a version of Bessel's inequality.) (ii) Now suppose that $f:{\mathbb R}\rightarrow{\mathbb C}$ is a continuously differentiable periodic function with period $2\pi$. If $f$ has derivative $g$, obtain a simple relation between $\hat{f}(n)$ and $\hat{g}(n)$. By applying the Cauchy-Schwarz inequality to $\sum_{n\neq 0,\ |n|\leq N} |n||\hat{g}(n)|$ show that \[\sum _{n\neq 0,\ |n|\leq N} |\hat{f}(n)| \leq A \frac{1}{2\pi}\int_{0}^{2\pi}|g(t)|^{2}\,dt,\] for some constant $A$. Conclude that $\sum |\hat{f}(n)|$ converges. (iii) Deduce, using Question~\ref{4.13}, that if $f:{\mathbb R}\rightarrow{\mathbb C}$ is a continuously differentiable periodic function with period $2\pi$ then \[f(t)=\sum_{n=-\infty}^{\infty}\hat{f}(n)\exp (int).\] \end{question} \begin{question} We continue with the ideas of Question~\ref{4.12}. From now on all sets will belong to the collection $\mathcal{K}$ of closed bounded non-empty subsets of ${\mathbb R}^{m}$. Define \[d(E,F)=\sup_{e\in E}d(e,F)+\sup_{f\in F}d(f,E).\] Show that $d$ is a metric on $\mathcal{K}$. Show also that $d$ is a complete metric. \end{question} \begin{question}\label{4.16} If $(X,d)$ is a complete metric space and $T:X\rightarrow X$ is a surjective map such that \[d(Tx,Ty)\geq Kd(x,y)\] for all $x,y\in X$ and some $K>1$ show that $T$ has a unique fixed point. By considering the map $T:{\mathbb R}\rightarrow{\mathbb R}$ defined by $T(x)=1+4n+2x$ for $0\leq x<1$ and $n$ an integer, or otherwise show that the condition $T$ surjective can not be dropped. \end{question} \begin{question}[A continuous nowhere differentiable function] The following construction is due to Van der Waerden. (i) Sketch the graph of the function $g$ given by the condition \[g(x+k)=|x|\ \ \ \ \mbox{if $k$ is any integer and $-1/2