# Analysis Ii

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Paper 1, Section II, E

commentLet $A \subset \mathbb{R}^{n}$ be an open subset. State what it means for a function $f: A \rightarrow \mathbb{R}^{m}$ to be differentiable at a point $p \in A$, and define its derivative $D f(p)$.

State and prove the chain rule for the derivative of $g \circ f$, where $g: \mathbb{R}^{m} \rightarrow \mathbb{R}^{r}$ is a differentiable function.

Let $M=M_{n}(\mathbb{R})$ be the vector space of $n \times n$ real-valued matrices, and $V \subset M$ the open subset consisting of all invertible ones. Let $f: V \rightarrow V$ be given by $f(A)=A^{-1}$.

(a) Show that $f$ is differentiable at the identity matrix, and calculate its derivative.

(b) For $C \in V$, let $l_{C}, r_{C}: M \rightarrow M$ be given by $l_{C}(A)=C A$ and $r_{C}(A)=A C$. Show that $r_{C} \circ f \circ l_{C}=f$ on $V$. Hence or otherwise, show that $f$ is differentiable at any point of $V$, and calculate $D f(C)(h)$ for $h \in M$.

Paper 2, Section I, E

commentConsider the map $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

$f(x, y)=\left(x^{1 / 3}+y^{2}, y^{5}\right)$

where $x^{1 / 3}$ denotes the unique real cube root of $x \in \mathbb{R}$.

(a) At what points is $f$ continuously differentiable? Calculate its derivative there.

(b) Show that $f$ has a local differentiable inverse near any $(x, y)$ with $x y \neq 0$.

You should justify your answers, stating accurately any results that you require.

Paper 2, Section II, 12E

comment(a) (i) Define what it means for two norms on a vector space to be Lipschitz equivalent.

(ii) Show that any two norms on a finite-dimensional vector space are Lipschitz equivalent.

(iii) Show that if two norms $\|\cdot\|,\|\cdot\|^{\prime}$ on a vector space $V$ are Lipschitz equivalent then the following holds: for any sequence $\left(v_{n}\right)$ in $V,\left(v_{n}\right)$ is Cauchy with respect to $\|\cdot\|$ if and only if it is Cauchy with respect to $\|\cdot\|^{\prime}$.

(b) Let $V$ be the vector space of real sequences $x=\left(x_{i}\right)$ such that $\sum\left|x_{i}\right|<\infty$. Let

$\|x\|_{\infty}=\sup \left\{\left|x_{i}\right|: i \in \mathbb{N}\right\},$

and for $1 \leqslant p<\infty$, let

$\|x\|_{p}=\left(\sum\left|x_{i}\right|^{p}\right)^{1 / p}$

You may assume that $\|\cdot\|_{\infty}$ and $\|\cdot\|_{p}$ are well-defined norms on $V$.

(i) Show that $\|\cdot\|_{p}$ is not Lipschitz equivalent to $\|\cdot\|_{\infty}$ for any $1 \leqslant p<\infty$.

(ii) Are there any $p, q$ with $1 \leqslant p<q<\infty$ such that $\|\cdot\|_{p}$ and $\|\cdot\|_{q}$ are Lipschitz equivalent? Justify your answer.

Paper 3, Section I, $2 E$

comment(a) Let $A \subset \mathbb{R}$. What does it mean for a function $f: A \rightarrow \mathbb{R}$ to be uniformly continuous?

(b) Which of the following functions are uniformly continuous? Briefly justify your answers.

(i) $f(x)=x^{2}$ on $\mathbb{R}$.

(ii) $f(x)=\sqrt{x}$ on $[0, \infty)$.

(iii) $f(x)=\cos (1 / x)$ on $[1, \infty)$.

Paper 3, Section II, E

comment(a) Carefully state the Picard-Lindelöf theorem on solutions to ordinary differential equations.

(b) Let $X=C\left([1, b], \mathbb{R}^{n}\right)$ be the set of continuous functions from a closed interval $[1, b]$ to $\mathbb{R}^{n}$, and let $\|\cdot\|$ be a norm on $\mathbb{R}^{n}$.

(i) Let $f \in X$. Show that for any $c \in[0, \infty)$ the norm

$\|f\|_{c}=\sup _{t \in[1, b]}\left\|f(t) t^{-c}\right\|$

is Lipschitz equivalent to the usual sup norm on $X$.

(ii) Assume that $F:[1, b] \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continuous and Lipschitz in the second variable, i.e. there exists $M>0$ such that

$\|F(t, x)-F(t, y)\| \leqslant M\|x-y\|$

for all $t \in[1, b]$ and all $x, y \in \mathbb{R}^{n}$. Define $\varphi: X \rightarrow X$ by

$\varphi(f)(t)=\int_{1}^{t} F(l, f(l)) d l$

for $t \in[1, b]$.

Show that there is a choice of $c$ such that $\varphi$ is a contraction on $\left(X,\|\cdot\|_{c}\right)$. Deduce that for any $y_{0} \in \mathbb{R}^{n}$, the differential equation

$D f(t)=F(t, f(t))$

has a unique solution on $[1, b]$ with $f(1)=y_{0}$.

Paper 4, Section I, E

commentLet $A \subset \mathbb{R}$. What does it mean to say that a sequence of real-valued functions on $A$ is uniformly convergent?

(i) If a sequence $\left(f_{n}\right)$ of real-valued functions on $A$ converges uniformly to $f$, and each $f_{n}$ is continuous, must $f$ also be continuous?

(ii) Let $f_{n}(x)=e^{-n x}$. Does the sequence $\left(f_{n}\right)$ converge uniformly on $[0,1]$ ?

(iii) If a sequence $\left(f_{n}\right)$ of real-valued functions on $[-1,1]$ converges uniformly to $f$, and each $f_{n}$ is differentiable, must $f$ also be differentiable?

Give a proof or counterexample in each case.

Paper 4, Section II, E

comment(a) (i) Show that a compact metric space must be complete.

(ii) If a metric space is complete and bounded, must it be compact? Give a proof or counterexample.

(b) A metric space $(X, d)$ is said to be totally bounded if for all $\epsilon>0$, there exists $N \in \mathbb{N}$ and $\left\{x_{1}, \ldots, x_{N}\right\} \subset X$ such that $X=\bigcup_{i=1}^{N} B_{\epsilon}\left(x_{i}\right) .$

(i) Show that a compact metric space is totally bounded.

(ii) Show that a complete, totally bounded metric space is compact.

[Hint: If $\left(x_{n}\right)$ is Cauchy, then there is a subsequence $\left(x_{n_{j}}\right)$ such that

$\left.\sum_{j} d\left(x_{n_{j+1}}, x_{n_{j}}\right)<\infty .\right]$

(iii) Consider the space $C[0,1]$ of continuous functions $f:[0,1] \rightarrow \mathbb{R}$, with the metric

$d(f, g)=\min \left\{\int_{0}^{1}|f(t)-g(t)| d t, 1\right\} .$

Is this space compact? Justify your answer.

Paper 1, Section II, F

commentLet $U \subset \mathbb{R}^{n}$ be a non-empty open set and let $f: U \rightarrow \mathbb{R}^{n}$.

(a) What does it mean to say that $f$ is differentiable? What does it mean to say that $f$ is a $C^{1}$ function?

If $f$ is differentiable, show that $f$ is continuous.

State the inverse function theorem.

(b) Suppose that $U$ is convex, $f$ is $C^{1}$ and that its derivative $D f(a)$ at a satisfies $\|D f(a)-I\|<1$ for all $a \in U$, where $I: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is the identity map and $\|\cdot\|$ denotes the operator norm. Show that $f$ is injective.

Explain why $f(U)$ is an open subset of $\mathbb{R}^{n}$.

Must it be true that $f(U)=\mathbb{R}^{n}$ ? What if $U=\mathbb{R}^{n}$ ? Give proofs or counter-examples as appropriate.

(c) Find the largest set $U \subset \mathbb{R}^{2}$ such that the map $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by $f(x, y)=\left(x^{2}-y^{2}, 2 x y\right)$ satisfies $\|D f(a)-I\|<1$ for every $a \in U$.

Paper 2, Section I, F

commentShow that $\|f\|_{1}=\int_{0}^{1}|f(x)| d x$ defines a norm on the space $C([0,1])$ of continuous functions $f:[0,1] \rightarrow \mathbb{R}$.

Let $\mathcal{S}$ be the set of continuous functions $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=g(1)=0$. Show that for each continuous function $f:[0,1] \rightarrow \mathbb{R}$, there is a sequence $g_{n} \in \mathcal{S}$ with $\sup _{x \in[0,1]}\left|g_{n}(x)\right| \leqslant \sup _{x \in[0,1]}|f(x)|$ such that $\left\|f-g_{n}\right\|_{1} \rightarrow 0$ as $n \rightarrow \infty$

Show that if $f:[0,1] \rightarrow \mathbb{R}$ is continuous and $\int_{0}^{1} f(x) g(x) d x=0$ for every $g \in \mathcal{S}$ then $f=0$.

Paper 2, Section II, F

comment(a) Let $(X, d)$ be a metric space, $A$ a non-empty subset of $X$ and $f: A \rightarrow \mathbb{R}$. Define what it means for $f$ to be Lipschitz. If $f$ is Lipschitz with Lipschitz constant $L$ and if

$F(x)=\inf _{y \in A}(f(y)+L d(x, y))$

for each $x \in X$, show that $F(x)=f(x)$ for each $x \in A$ and that $F: X \rightarrow \mathbb{R}$ is Lipschitz with Lipschitz constant $L$. (Be sure to justify that $F(x) \in \mathbb{R}$, i.e. that the infimum is finite for every $x \in X$.)

(b) What does it mean to say that two norms on a vector space are Lipschitz equivalent?

Let $V$ be an $n$-dimensional real vector space equipped with a norm $\|$. Let $\left\{e_{1}, e_{2}, \ldots, e_{n}\right\}$ be a basis for $V$. Show that the map $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ defined by $g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left\|x_{1} e_{1}+x_{2} e_{2}+\ldots+x_{n} e_{n}\right\|$ is continuous. Deduce that any two norms on $V$ are Lipschitz equivalent.

(c) Prove that for each positive integer $n$ and each $a \in(0,1]$, there is a constant $C>0$ with the following property: for every polynomial $p$ of degree $\leqslant n$, there is a point $y \in[0, a]$ such that

$\sup _{x \in[0,1]}\left|p^{\prime}(x)\right| \leqslant C|p(y)|$

where $p^{\prime}$ is the derivative of $p$.

Paper 3, Section I, $2 F$

commentFor a continuous function $f=\left(f_{1}, f_{2}, \ldots, f_{m}\right):[0,1] \rightarrow \mathbb{R}^{m}$, define

$\int_{0}^{1} f(t) d t=\left(\int_{0}^{1} f_{1}(t) d t, \int_{0}^{1} f_{2}(t) d t, \ldots, \int_{0}^{1} f_{m}(t) d t\right)$

Show that

$\left\|\int_{0}^{1} f(t) d t\right\|_{2} \leqslant \int_{0}^{1}\|f(t)\|_{2} d t$

for every continuous function $f:[0,1] \rightarrow \mathbb{R}^{m}$, where $\|\cdot\|_{2}$ denotes the Euclidean norm on $\mathbb{R}^{m}$.

Find all continuous functions $f:[0,1] \rightarrow \mathbb{R}^{m}$ with the property that

$\left\|\int_{0}^{1} f(t) d t\right\|=\int_{0}^{1}\|f(t)\| d t$

regardless of the norm $\|\cdot\|$ on $\mathbb{R}^{m}$.

[Hint: start by analysing the case when $\|\cdot\|$ is the Euclidean norm $\|\cdot\|_{2}$.]

Paper 3, Section II, F

comment(a) Let $A \subset \mathbb{R}^{m}$ and let $f, f_{n}: A \rightarrow \mathbb{R}$ be functions for $n=1,2,3, \ldots$ What does it mean to say that the sequence $\left(f_{n}\right)$ converges uniformly to $f$ on $A$ ? What does it mean to say that $f$ is uniformly continuous?

(b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a uniformly continuous function. Determine whether each of the following statements is true or false. Give reasons for your answers.

(i) If $f_{n}(x)=f(x+1 / n)$ for each $n=1,2,3, \ldots$ and each $x \in \mathbb{R}$, then $f_{n} \rightarrow f$ uniformly on $\mathbb{R}$.

(ii) If $g_{n}(x)=(f(x+1 / n))^{2}$ for each $n=1,2,3, \ldots$ and each $x \in \mathbb{R}$, then $g_{n} \rightarrow(f)^{2}$ uniformly on $\mathbb{R}$.

(c) Let $A$ be a closed, bounded subset of $\mathbb{R}^{m}$. For each $n=1,2,3, \ldots$, let $g_{n}: A \rightarrow \mathbb{R}$ be a continuous function such that $\left(g_{n}(x)\right)$ is a decreasing sequence for each $x \in A$. If $\delta \in \mathbb{R}$ is such that for each $n$ there is $x_{n} \in A$ with $g_{n}\left(x_{n}\right) \geqslant \delta$, show that there is $x_{0} \in A$ such that $\lim _{n \rightarrow \infty} g_{n}\left(x_{0}\right) \geqslant \delta$.

Deduce the following: If $f_{n}: A \rightarrow \mathbb{R}$ is a continuous function for each $n=1,2,3, \ldots$ such that $\left(f_{n}(x)\right)$ is a decreasing sequence for each $x \in A$, and if the pointwise limit of $\left(f_{n}\right)$ is a continuous function $f: A \rightarrow \mathbb{R}$, then $f_{n} \rightarrow f$ uniformly on $A$.

Paper 4, Section I, F

commentState the Bolzano-Weierstrass theorem in $\mathbb{R}$. Use it to deduce the BolzanoWeierstrass theorem in $\mathbb{R}^{n}$.

Let $D$ be a closed, bounded subset of $\mathbb{R}^{n}$, and let $f: D \rightarrow \mathbb{R}$ be a function. Let $\mathcal{S}$ be the set of points in $D$ where $f$ is discontinuous. For $\rho>0$ and $z \in \mathbb{R}^{n}$, let $B_{\rho}(z)$ denote the ball $\left\{x \in \mathbb{R}^{n}:\|x-z\|<\rho\right\}$. Prove that for every $\epsilon>0$, there exists $\delta>0$ such that $|f(x)-f(y)|<\epsilon$ whenever $x \in D, y \in D \backslash \cup_{z \in \mathcal{S}} B_{\epsilon}(z)$ and $\|x-y\|<\delta$.

(If you use the fact that a continuous function on a compact metric space is uniformly continuous, you must prove it.)

Paper 4, Section II, F

comment(a) Define what it means for a metric space $(X, d)$ to be complete. Give a metric $d$ on the interval $I=(0,1]$ such that $(I, d)$ is complete and such that a subset of $I$ is open with respect to $d$ if and only if it is open with respect to the Euclidean metric on $I$. Be sure to prove that $d$ has the required properties.

(b) Let $(X, d)$ be a complete metric space.

(i) If $Y \subset X$, show that $Y$ taken with the subspace metric is complete if and only if $Y$ is closed in $X$.

(ii) Let $f: X \rightarrow X$ and suppose that there is a number $\lambda \in(0,1)$ such that $d(f(x), f(y)) \leqslant \lambda d(x, y)$ for every $x, y \in X$. Show that there is a unique point $x_{0} \in X$ such that $f\left(x_{0}\right)=x_{0}$.

Deduce that if $\left(a_{n}\right)$ is a sequence of points in $X$ converging to a point $a \neq x_{0}$, then there are integers $\ell$ and $m \geqslant \ell$ such that $f\left(a_{m}\right) \neq a_{n}$ for every $n \geqslant \ell$.

Paper 1, Section II, G

commentWhat does it mean to say that a real-valued function on a metric space is uniformly continuous? Show that a continuous function on a closed interval in $\mathbb{R}$ is uniformly continuous.

What does it mean to say that a real-valued function on a metric space is Lipschitz? Show that if a function is Lipschitz then it is uniformly continuous.

Which of the following statements concerning continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ are true and which are false? Justify your answers.

(i) If $f$ is bounded then $f$ is uniformly continuous.

(ii) If $f$ is differentiable and $f^{\prime}$ is bounded, then $f$ is uniformly continuous.

(iii) There exists a sequence of uniformly continuous functions converging pointwise to $f$.

Paper 2, Section I, G

commentLet $X \subset \mathbb{R}$. What does it mean to say that a sequence of real-valued functions on $X$ is uniformly convergent?

Let $f, f_{n}(n \geqslant 1): \mathbb{R} \rightarrow \mathbb{R}$ be functions.

(a) Show that if each $f_{n}$ is continuous, and $\left(f_{n}\right)$ converges uniformly on $\mathbb{R}$ to $f$, then $f$ is also continuous.

(b) Suppose that, for every $M>0,\left(f_{n}\right)$ converges uniformly on $[-M, M]$. Need $\left(f_{n}\right)$ converge uniformly on $\mathbb{R}$ ? Justify your answer.

Paper 2, Section II, G

commentLet $V$ be a real vector space. What is a norm on $V$ ? Show that if $\|-\|$ is a norm on $V$, then the maps $T_{v}: x \mapsto x+v\left(\right.$ for $v \in V$ ) and $m_{a}: x \mapsto a x$ (for $a \in \mathbb{R}$ ) are continuous with respect to the norm.

Let $B \subset V$ be a subset containing 0 . Show that there exists at most one norm on $V$ for which $B$ is the open unit ball.

Suppose that $B$ satisfies the following two properties:

if $v \in V$ is a nonzero vector, then the line $\mathbb{R} v \subset V$ meets $B$ in a set of the form $\{t v:-\lambda<t<\lambda\}$ for some $\lambda>0$;

if $x, y \in B$ and $s, t>0$ then $(s+t)^{-1}(s x+t y) \in B$.

Show that there exists a norm $\|-\|_{B}$ for which $B$ is the open unit ball.

Identify $\|-\|_{B}$ in the following two cases:

(i) $V=\mathbb{R}^{n}, B=\left\{\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}:-1<x_{i}<1\right.$ for all $\left.i\right\}$.

(ii) $V=\mathbb{R}^{2}, B$ the interior of the square with vertices $(\pm 1,0),(0, \pm 1)$.

Let $C \subset \mathbb{R}^{2}$ be the set

$C=\left\{\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}:\left|x_{1}\right|<1,\left|x_{2}\right|<1, \text { and }\left(\left|x_{1}\right|-1\right)^{2}+\left(\left|x_{2}\right|-1\right)^{2}>1\right\}$

Is there a norm on $\mathbb{R}^{2}$ for which $C$ is the open unit ball? Justify your answer.

Paper 3, Section I, G

commentWhat does it mean to say that a metric space is complete? Which of the following metric spaces are complete? Briefly justify your answers.

(i) $[0,1]$ with the Euclidean metric.

(ii) $\mathbb{Q}$ with the Euclidean metric.

(iii) The subset

$\{(0,0)\} \cup\{(x, \sin (1 / x)) \mid x>0\} \subset \mathbb{R}^{2}$

with the metric induced from the Euclidean metric on $\mathbb{R}^{2}$.

Write down a metric on $\mathbb{R}$ with respect to which $\mathbb{R}$ is not complete, justifying your answer.

[You may assume throughout that $\mathbb{R}$ is complete with respect to the Euclidean metric.

Paper 3, Section II, G

commentWhat is a contraction map on a metric space $X$ ? State and prove the contraction mapping theorem.

Let $(X, d)$ be a complete non-empty metric space. Show that if $f: X \rightarrow X$ is a map for which some iterate $f^{k}(k \geqslant 1)$ is a contraction map, then $f$ has a unique fixed point. Show that $f$ itself need not be a contraction map.

Let $f:[0, \infty) \rightarrow[0, \infty)$ be the function

$f(x)=\frac{1}{3}\left(x+\sin x+\frac{1}{x+1}\right)$

Show that $f$ has a unique fixed point.

Paper 4, Section I, G

commentState the chain rule for the composition of two differentiable functions $f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{p}$.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be differentiable. For $c \in \mathbb{R}$, let $g(x)=f(x, c-x)$. Compute the derivative of $g$. Show that if $\partial f / \partial x=\partial f / \partial y$ throughout $\mathbb{R}^{2}$, then $f(x, y)=h(x+y)$ for some function $h: \mathbb{R} \rightarrow \mathbb{R}$.

Paper 4, Section II, G

commentLet $U \subset \mathbb{R}^{m}$ be a nonempty open set. What does it mean to say that a function $f: U \rightarrow \mathbb{R}^{n}$ is differentiable?

Let $f: U \rightarrow \mathbb{R}$ be a function, where $U \subset \mathbb{R}^{2}$ is open. Show that if the first partial derivatives of $f$ exist and are continuous on $U$, then $f$ is differentiable on $U$.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the function

$f(x, y)= \begin{cases}0 & (x, y)=(0,0) \\ \frac{x^{3}+2 y^{4}}{x^{2}+y^{2}} & (x, y) \neq(0,0)\end{cases}$

Determine, with proof, where $f$ is differentiable.

Paper 1, Section II, G

commentLet $(X, d)$ be a metric space.

(a) What does it mean to say that $\left(x_{n}\right)_{n}$ is a Cauchy sequence in $X$ ? Show that if $\left(x_{n}\right)_{n}$ is a Cauchy sequence, then it converges if it contains a convergent subsequence.

(b) Let $\left(x_{n}\right)_{n}$ be a Cauchy sequence in $X$.

(i) Show that for every $m \geqslant 1$, the sequence $\left(d\left(x_{m}, x_{n}\right)\right)_{n}$ converges to some $d_{m} \in \mathbb{R}$.

(ii) Show that $d_{m} \rightarrow 0$ as $m \rightarrow \infty$.

(iii) Let $\left(y_{n}\right)_{n}$ be a subsequence of $\left(x_{n}\right)_{n}$. If $\ell, m$ are such that $y_{\ell}=x_{m}$, show that $d\left(y_{\ell}, y_{n}\right) \rightarrow d_{m}$ as $n \rightarrow \infty$.

(iv) Show also that for every $m$ and $n$,

$d_{m}-d_{n} \leqslant d\left(x_{m}, x_{n}\right) \leqslant d_{m}+d_{n}$

(v) Deduce that $\left(x_{n}\right)_{n}$ has a subsequence $\left(y_{n}\right)_{n}$ such that for every $m$ and $n$,

$d\left(y_{m+1}, y_{m}\right) \leqslant \frac{1}{3} d\left(y_{m}, y_{m-1}\right)$

and

$d\left(y_{m+1}, y_{n+1}\right) \leqslant \frac{1}{2} d\left(y_{m}, y_{n}\right)$

(c) Suppose that every closed subset $Y$ of $X$ has the property that every contraction mapping $Y \rightarrow Y$ has a fixed point. Prove that $X$ is complete.

Paper 2, Section I, G

comment(a) What does it mean to say that the function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is differentiable at the point $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$ ? Show from your definition that if $f$ is differentiable at $x$, then $f$ is continuous at $x$.

(b) Suppose that there are functions $g_{j}: \mathbb{R} \rightarrow \mathbb{R}^{m}(1 \leqslant j \leqslant n)$ such that for every $x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$,

$f(x)=\sum_{j=1}^{n} g_{j}\left(x_{j}\right) .$

Show that $f$ is differentiable at $x$ if and only if each $g_{j}$ is differentiable at $x_{j}$.

(c) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by

$f(x, y)=|x|^{3 / 2}+|y|^{1 / 2}$

Determine at which points $(x, y) \in \mathbb{R}^{2}$ the function $f$ is differentiable.

Paper 2, Section II, G

comment(a) What is a norm on a real vector space?

(b) Let $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$ be the space of linear maps from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$. Show that

$\|A\|=\sup _{0 \neq x \in \mathbb{R}^{m}} \frac{\|A x\|}{\|x\|}, \quad A \in L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right),$

defines a norm on $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$, and that if $B \in L\left(\mathbb{R}^{\ell}, \mathbb{R}^{m}\right)$ then $\|A B\| \leqslant\|A\|\|B\|$.

(c) Let $M_{n}$ be the space of $n \times n$ real matrices, identified with $L\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right)$ in the usual way. Let $U \subset M_{n}$ be the subset

$U=\left\{X \in M_{n} \mid I-X \text { is invertible }\right\}$

Show that $U$ is an open subset of $M_{n}$ which contains the set $V=\left\{X \in M_{n} \mid\|X\|<1\right\}$.

(d) Let $f: U \rightarrow M_{n}$ be the map $f(X)=(I-X)^{-1}$. Show carefully that the series $\sum_{k=0}^{\infty} X^{k}$ converges on $V$ to $f(X)$. Hence or otherwise, show that $f$ is twice differentiable at 0 , and compute its first and second derivatives there.

Paper 3, Section I, G

comment(a) Let $X$ be a subset of $\mathbb{R}$. What does it mean to say that a sequence of functions $f_{n}: X \rightarrow \mathbb{R}(n \in \mathbb{N})$ is uniformly convergent?

(b) Which of the following sequences of functions are uniformly convergent? Justify your answers.

(i) $f_{n}:(0,1) \rightarrow \mathbb{R}, \quad f_{n}(x)=\frac{1-x^{n}}{1-x}$

(ii) $f_{n}:(0,1) \rightarrow \mathbb{R}, \quad f_{n}(x)=\sum_{k=1}^{n} \frac{1}{k^{2}} x^{k}$.

(iii) $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$, $\quad f_{n}(x)=x / n$.

(iv) $f_{n}:[0, \infty) \rightarrow \mathbb{R}, \quad f_{n}(x)=x e^{-n x}$.

Paper 3, Section II, G

commentLet $X$ be a metric space.

(a) What does it mean to say that a function $f: X \rightarrow \mathbb{R}$ is uniformly continuous? What does it mean to say that $f$ is Lipschitz? Show that if $f$ is Lipschitz then it is uniformly continuous. Show also that if $\left(x_{n}\right)_{n}$ is a Cauchy sequence in $X$, and $f$ is uniformly continuous, then the sequence $\left(f\left(x_{n}\right)\right)_{n}$ is convergent.

(b) Let $f: X \rightarrow \mathbb{R}$ be continuous, and $X$ be sequentially compact. Show that $f$ is uniformly continuous. Is $f$ necessarily Lipschitz? Justify your answer.

(c) Let $Y$ be a dense subset of $X$, and let $g: Y \rightarrow \mathbb{R}$ be a continuous function. Show that there exists at most one continuous function $f: X \rightarrow \mathbb{R}$ such that for all $y \in Y$, $f(y)=g(y)$. Prove that if $g$ is uniformly continuous, then such a function $f$ exists, and is uniformly continuous.

[A subset $Y \subset X$ is dense if for any nonempty open subset $U \subset X$, the intersection $U \cap Y$ is nonempty.]

Paper 4, Section I, G

comment(a) What does it mean to say that a mapping $f: X \rightarrow X$ from a metric space to itself is a contraction?

(b) State carefully the contraction mapping theorem.

(c) Let $\left(a_{1}, a_{2}, a_{3}\right) \in \mathbb{R}^{3}$. By considering the metric space $\left(\mathbb{R}^{3}, d\right)$ with

$d(x, y)=\sum_{i=1}^{3}\left|x_{i}-y_{i}\right|$

or otherwise, show that there exists a unique solution $\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}$ of the system of equations

$\begin{aligned} &x_{1}=a_{1}+\frac{1}{6}\left(\sin x_{2}+\sin x_{3}\right), \\ &x_{2}=a_{2}+\frac{1}{6}\left(\sin x_{1}+\sin x_{3}\right), \\ &x_{3}=a_{3}+\frac{1}{6}\left(\sin x_{1}+\sin x_{2}\right) . \end{aligned}$

Paper 4, Section II, G

comment(a) Let $V$ be a real vector space. What does it mean to say that two norms on $V$ are Lipschitz equivalent? Prove that every norm on $\mathbb{R}^{n}$ is Lipschitz equivalent to the Euclidean norm. Hence or otherwise, show that any linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ is continuous.

(b) Let $f: U \rightarrow V$ be a linear map between normed real vector spaces. We say that $f$ is bounded if there exists a constant $C$ such that for all $u \in U,\|f(u)\| \leqslant C\|u\|$. Show that $f$ is bounded if and only if $f$ is continuous.

(c) Let $\ell^{2}$ denote the space of sequences $\left(x_{n}\right)_{n \geqslant 1}$ of real numbers such that $\sum_{n \geqslant 1} x_{n}^{2}$ is convergent, with the norm $\left\|\left(x_{n}\right)_{n}\right\|=\left(\sum_{n \geqslant 1} x_{n}^{2}\right)^{1 / 2}$. Let $e_{m} \in \ell^{2}$ be the sequence $e_{m}=\left(x_{n}\right)_{n}$ with $x_{m}=1$ and $x_{n}=0$ if $n \neq m$. Let $w$ be the sequence $\left(2^{-n}\right)_{n}$. Show that the subset $\{w\} \cup\left\{e_{m} \mid m \geqslant 1\right\}$ is linearly independent. Let $V \subset \ell^{2}$ be the subspace it spans, and consider the linear map $f: V \rightarrow \mathbb{R}$ defined by

$f(w)=1, \quad f\left(e_{m}\right)=0 \quad \text { for all } m \geqslant 1 .$

Is $f$ continuous? Justify your answer.

Paper 1, Section II, G

commentDefine what it means for a sequence of functions $f_{n}:[0,1] \rightarrow \mathbb{R}$ to converge uniformly on $[0,1]$ to a function $f$.

Let $f_{n}(x)=n^{p} x e^{-n^{q} x}$, where $p, q$ are positive constants. Determine all the values of $(p, q)$ for which $f_{n}(x)$ converges pointwise on $[0,1]$. Determine all the values of $(p, q)$ for which $f_{n}(x)$ converges uniformly on $[0,1]$.

Let now $f_{n}(x)=e^{-n x^{2}}$. Determine whether or not $f_{n}$ converges uniformly on $[0,1]$.

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. Show that the sequence $x^{n} f(x)$ is uniformly convergent on $[0,1]$ if and only if $f(1)=0$.

[If you use any theorems about uniform convergence, you should prove these.]

Paper 2, Section I, G

commentShow that the map $f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by

$f(x, y, z)=\left(x-y-z, x^{2}+y^{2}+z^{2}, x y z\right)$

is differentiable everywhere and find its derivative.

Stating accurately any theorem that you require, show that $f$ has a differentiable local inverse at a point $(x, y, z)$ if and only if

$(x+y)(x+z)(y-z) \neq 0 .$

$\begin{aligned} & p(f)=\sup |f|, \quad q(f)=\sup \left(|f|+\left|f^{\prime}\right|\right), \\ & r(f)=\sup \left|f^{\prime}\right|, \quad s(f)=\left|\int_{-1}^{1} f(x) d x\right| \end{aligned}$

Paper 2, Section II, G

commentLet $E, F$ be normed spaces with norms $\|\cdot\|_{E},\|\cdot\|_{F}$. Show that for a map $f: E \rightarrow F$ and $a \in E$, the following two statements are equivalent:

(i) For every given $\varepsilon>0$ there exists $\delta>0$ such that $\|f(x)-f(a)\|_{F}<\varepsilon$ whenever $\|x-a\|_{E}<\delta$

(ii) $f\left(x_{n}\right) \rightarrow f(a)$ for each sequence $x_{n} \rightarrow a$.

We say that $f$ is continuous at $a$ if (i), or equivalently (ii), holds.

Let now $\left(E,\|\cdot\|_{E}\right)$ be a normed space. Let $A \subset E$ be a non-empty closed subset and define $d(x, A)=\inf \left\{\|x-a\|_{E}: a \in A\right\}$. Show that

$|d(x, A)-d(y, A)| \leqslant\|x-y\|_{E} \text { for all } x, y \in E .$

In the case when $E=\mathbb{R}^{n}$ with the standard Euclidean norm, show that there exists $a \in A$ such that $d(x, A)=\|x-a\|$.

Let $A, B$ be two disjoint closed sets in $\mathbb{R}^{n}$. Must there exist disjoint open sets $U, V$ such that $A \subset U$ and $B \subset V$ ? Must there exist $a \in A$ and $b \in B$ such that $d(a, b) \leqslant d(x, y)$ for all $x \in A$ and $y \in B$ ? For each answer, give a proof or counterexample as appropriate.

Paper 3, Section I, G

commentDefine what is meant by a uniformly continuous function $f$ on a subset $E$ of a metric space. Show that every continuous function on a closed, bounded interval is uniformly continuous. [You may assume the Bolzano-Weierstrass theorem.]

Suppose that a function $g:[0, \infty) \rightarrow \mathbb{R}$ is continuous and tends to a finite limit at $\infty$. Is $g$ necessarily uniformly continuous on $[0, \infty) ?$ Give a proof or a counterexample as appropriate.

Paper 3, Section II, G

commentDefine what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ to be differentiable at $x \in \mathbb{R}^{n}$ with derivative $D f(x)$.

State and prove the chain rule for the derivative of $g \circ f$, where $g: \mathbb{R}^{m} \rightarrow \mathbb{R}^{p}$ is a differentiable function.

Now let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a differentiable function and let $g(x)=f(x, c-x)$ where $c$ is a constant. Show that $g$ is differentiable and find its derivative in terms of the partial derivatives of $f$. Show that if $D_{1} f(x, y)=D_{2} f(x, y)$ holds everywhere in $\mathbb{R}^{2}$, then $f(x, y)=h(x+y)$ for some differentiable function $h .$

Paper 4, Section I, G

commentDefine what is meant for two norms on a vector space to be Lipschitz equivalent.

Let $C_{c}^{1}([-1,1])$ denote the vector space of continuous functions $f:[-1,1] \rightarrow \mathbb{R}$ with continuous first derivatives and such that $f(x)=0$ for $x$ in some neighbourhood of the end-points $-1$ and 1 . Which of the following four functions $C_{c}^{1}([-1,1]) \rightarrow \mathbb{R}$ define norms on $C_{c}^{1}([-1,1])$ (give a brief explanation)?

Among those that define norms, which pairs are Lipschitz equivalent? Justify your answer.

Paper 4, Section II, G

commentConsider the space $\ell^{\infty}$ of bounded real sequences $x=\left(x_{i}\right)_{i=1}^{\infty}$ with the norm $\|x\|_{\infty}=\sup _{i}\left|x_{i}\right|$. Show that for every bounded sequence $x^{(n)}$ in $\ell^{\infty}$ there is a subsequence $x^{\left(n_{j}\right)}$ which converges in every coordinate, i.e. the sequence $\left(x_{i}^{\left(n_{j}\right)}\right)_{j=1}^{\infty}$ of real numbers converges for each $i$. Does every bounded sequence in $\ell^{\infty}$ have a convergent subsequence? Justify your answer.

Let $\ell^{1} \subset \ell^{\infty}$ be the subspace of real sequences $x=\left(x_{i}\right)_{i=1}^{\infty}$ such that $\sum_{i=1}^{\infty}\left|x_{i}\right|$ converges. Is $\ell^{1}$ complete in the norm $\|\cdot\|_{\infty}$ (restricted from $\ell^{\infty}$ to $\left.\ell^{1}\right)$ ? Justify your answer.

Suppose that $\left(x_{i}\right)$ is a real sequence such that, for every $\left(y_{i}\right) \in \ell^{\infty}$, the series $\sum_{i=1}^{\infty} x_{i} y_{i}$ converges. Show that $\left(x_{i}\right) \in \ell^{1} .$

Suppose now that $\left(x_{i}\right)$ is a real sequence such that, for every $\left(y_{i}\right) \in \ell^{1}$, the series $\sum_{i=1}^{\infty} x_{i} y_{i}$ converges. Show that $\left(x_{i}\right) \in \ell^{\infty} .$

Paper 1, Section II, F

commentDefine what it means for two norms on a real vector space $V$ to be Lipschitz equivalent. Show that if two norms on $V$ are Lipschitz equivalent and $F \subset V$, then $F$ is closed in one norm if and only if $F$ is closed in the other norm.

Show that if $V$ is finite-dimensional, then any two norms on $V$ are Lipschitz equivalent.

Show that $\|f\|_{1}=\int_{0}^{1}|f(x)| d x$ is a norm on the space $C[0,1]$ of continuous realvalued functions on $[0,1]$. Is the set $S=\{f \in C[0,1]: f(1 / 2)=0\}$ closed in the norm $\|\cdot\| 1$ ?

Determine whether or not the norm $\|\cdot\|_{1}$ is Lipschitz equivalent to the uniform $\operatorname{norm}\|\cdot\|_{\infty}$ on $C[0,1]$.

[You may assume the Bolzano-Weierstrass theorem for sequences in $\mathbb{R}^{n}$.]

Paper 2, Section I, F

commentDefine what is meant by a uniformly continuous function on a set $E \subset \mathbb{R}$.

If $f$ and $g$ are uniformly continuous functions on $\mathbb{R}$, is the (pointwise) product $f g$ necessarily uniformly continuous on $\mathbb{R}$ ?

Is a uniformly continuous function on $(0,1)$ necessarily bounded?

Is $\cos (1 / x)$ uniformly continuous on $(0,1) ?$

Justify your answers.

Paper 2, Section II, $12 \mathrm{~F}$

commentLet $X, Y$ be subsets of $\mathbb{R}^{n}$ and define $X+Y=\{x+y: x \in X, y \in Y\}$. For each of the following statements give a proof or a counterexample (with justification) as appropriate.

(i) If each of $X, Y$ is bounded and closed, then $X+Y$ is bounded and closed.

(ii) If $X$ is bounded and closed and $Y$ is closed, then $X+Y$ is closed.

(iii) If $X, Y$ are both closed, then $X+Y$ is closed.

(iv) If $X$ is open and $Y$ is closed, then $X+Y$ is open.

[The Bolzano-Weierstrass theorem in $\mathbb{R}^{n}$ may be assumed without proof.]

Paper 3, Section I, F

commentLet $U \subset \mathbb{R}^{n}$ be an open set and let $f: U \rightarrow \mathbb{R}$ be a differentiable function on $U$ such that $\left\|\left.D f\right|_{x}\right\| \leqslant M$ for some constant $M$ and all $x \in U$, where $\left\|\left.D f\right|_{x}\right\|$ denotes the operator norm of the linear map $\left.D f\right|_{x}$. Let $[a, b]=\{t a+(1-t) b: 0 \leqslant t \leqslant 1\}\left(a, b, \in \mathbb{R}^{n}\right)$ be a straight-line segment contained in $U$. Prove that $|f(b)-f(a)| \leqslant M\|b-a\|$, where $\|\cdot\|$ denotes the Euclidean norm on $\mathbb{R}^{n}$.

Prove that if $U$ is an open ball and $\left.D f\right|_{x}=0$ for each $x \in U$, then $f$ is constant on $U$.

Paper 3, Section II, F

commentLet $f_{n}, n=1,2, \ldots$, be continuous functions on an open interval $(a, b)$. Prove that if the sequence $\left(f_{n}\right)$ converges to $f$ uniformly on $(a, b)$ then the function $f$ is continuous on $(a, b)$.

If instead $\left(f_{n}\right)$ is only known to converge pointwise to $f$ and $f$ is continuous, must $\left(f_{n}\right)$ be uniformly convergent? Justify your answer.

Suppose that a function $f$ has a continuous derivative on $(a, b)$ and let

$g_{n}(x)=n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)$

Stating clearly any standard results that you require, show that the functions $g_{n}$ converge uniformly to $f^{\prime}$ on each interval $[\alpha, \beta] \subset(a, b)$.

Paper 4, Section I, F

commentDefine a contraction mapping and state the contraction mapping theorem.

Let $C[0,1]$ be the space of continuous real-valued functions on $[0,1]$ endowed with the uniform norm. Show that the map $A: C[0,1] \rightarrow C[0,1]$ defined by

$A f(x)=\int_{0}^{x} f(t) d t$

is not a contraction mapping, but that $A \circ A$ is.

Paper 4, Section II, F

commentLet $U \subset \mathbb{R}^{2}$ be an open set. Define what it means for a function $f: U \rightarrow \mathbb{R}$ to be differentiable at a point $\left(x_{0}, y_{0}\right) \in U$.

Prove that if the partial derivatives $D_{1} f$ and $D_{2} f$ exist on $U$ and are continuous at $\left(x_{0}, y_{0}\right)$, then $f$ is differentiable at $\left(x_{0}, y_{0}\right)$.

If $f$ is differentiable on $U$ must $D_{1} f, D_{2} f$ be continuous at $\left(x_{0}, y_{0}\right) ?$ Give a proof or counterexample as appropriate.

The function $h: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is defined by

$h(x, y)=x y \sin (1 / x) \quad \text { for } x \neq 0, \quad h(0, y)=0$

Determine all the points $(x, y)$ at which $h$ is differentiable.

Paper 1, Section II, F

commentDefine what it means for a sequence of functions $k_{n}: A \rightarrow \mathbb{R}, n=1,2, \ldots$, to converge uniformly on an interval $A \subset \mathbb{R}$.

By considering the functions $k_{n}(x)=\frac{\sin (n x)}{\sqrt{n}}$, or otherwise, show that uniform convergence of a sequence of differentiable functions does not imply uniform convergence of their derivatives.

Now suppose $k_{n}(x)$ is continuously differentiable on $A$ for each $n$, that $k_{n}\left(x_{0}\right)$ converges as $n \rightarrow \infty$ for some $x_{0} \in A$, and moreover that the derivatives $k_{n}^{\prime}(x)$ converge uniformly on $A$. Prove that $k_{n}(x)$ converges to a continuously differentiable function $k(x)$ on $A$, and that

$k^{\prime}(x)=\lim _{n \rightarrow \infty} k_{n}^{\prime}(x)$

Hence, or otherwise, prove that the function

$\sum_{n=1}^{\infty} \frac{x^{n} \sin (n x)}{n^{3}+1}$

is continuously differentiable on $(-1,1)$.

Paper 2, Section I, F

commentLet $\mathcal{C}[a, b]$ denote the vector space of continuous real-valued functions on the interval $[a, b]$, and let $\mathcal{C}^{\prime}[a, b]$ denote the subspace of continuously differentiable functions.

Show that $\|f\|_{1}=\max |f|+\max \left|f^{\prime}\right|$ defines a norm on $\mathcal{C}^{\prime}[a, b]$. Show furthermore that the map $\Phi: f \mapsto f^{\prime}((a+b) / 2)$ takes the closed unit ball $\left\{\|f\|_{1} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$.

If instead we had used the norm $\|f\|_{0}=\max |f|$ restricted from $\mathcal{C}[a, b]$ to $\mathcal{C}^{\prime}[a, b]$, would $\Phi$ take the closed unit ball $\left\{\|f\|_{0} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$ ? Justify your answer.

Paper 2, Section II, F

Let $f: U \rightarrow \mathbb{R}$ be continuous on an open set $U \subset \mathbb{R}^{2}$. Suppose that on $U$ the partial derivatives $D_{1} f, D_{2} f, D_{1} D_{2} f$ and $D_{2} D_{1} f$ exist and are continuous. Prove that $D_{1} D_{2} f=D_{2} D_{1} f$