• # A1.3

(i) Let $H$ be a Hilbert space, and let $M$ be a non-zero closed vector subspace of $H$. For $x \in H$, show that there is a unique closest point $P_{M}(x)$ to $x$ in $M$.

(ii) (a) Let $x \in H$. Show that $x-P_{M}(x) \in M^{\perp}$. Show also that if $y \in M$ and $x-y \in M^{\perp}$ then $y=P_{M}(x)$.

(b) Deduce that $H=M \bigoplus M^{\perp}$.

(c) Show that the map $P_{M}$ from $H$ to $M$ is a continuous linear map, with $\left\|P_{M}\right\|=1$.

(d) Show that $P_{M}$ is the projection onto $M$ along $M^{\perp}$.

Now suppose that $A$ is a subspace of $H$ that is not necessarily closed. Explain why $A^{\perp}=\{0\}$ implies that $A$ is dense in $H .$

Give an example of a subspace of $l^{2}$ that is dense in $l^{2}$ but is not equal to $l^{2}$.

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• # A2.3 B2.2

(i) Prove Riesz's Lemma, that if $V$ is a normed space and $A$ is a vector subspace of $V$ such that for some $0 \leqslant k<1$ we have $d(x, A) \leqslant k$ for all $x \in V$ with $\|x\|=1$, then $A$ is dense in $V$. [Here $d(x, A)$ denotes the distance from $x$ to $A$.]

Deduce that any normed space whose unit ball is compact is finite-dimensional. [You may assume that every finite-dimensional normed space is complete.]

Give an example of a sequence $f_{1}, f_{2}, \ldots$ in an infinite-dimensional normed space such that $\left\|f_{n}\right\| \leqslant 1$ for all $n$, but $f_{1}, f_{2}, \ldots$ has no convergent subsequence.

(ii) Let $V$ be a vector space, and let $\|\cdot\|_{1}$ and $\|.\|_{2}$ be two norms on $V$. What does it mean to say that $\|\cdot\|_{1}$ and $\|.\|_{2}$ are equivalent?

Show that on a finite-dimensional vector space all norms are equivalent. Deduce that every finite-dimensional normed space is complete.

Exhibit two norms on the vector space $l^{1}$ that are not equivalent.

In addition, exhibit two norms on the vector space $l^{\infty}$ that are not equivalent.

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• # A3.3 B3.2

(i) Let $H$ be an infinite-dimensional Hilbert space. Show that $H$ has a (countable) orthonormal basis if and only if $H$ has a countable dense subset. [You may assume familiarity with the Gram-Schmidt process.]

State and prove Bessel's inequality.

(ii) State Parseval's equation. Using this, prove that if $H$ has a countable dense subset then there is a surjective isometry from $H$ to $l^{2}$.

Explain carefully why the functions $e^{i n \theta}, n \in \mathbb{Z}$, form an orthonormal basis for $L^{2}(\mathbb{T})$

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• # A4.3

State and prove the Dominated Convergence Theorem. [You may assume the Monotone Convergence Theorem.]

Let $a$ and $p$ be real numbers, with $a>0$. Prove carefully that

$\int_{0}^{\infty} e^{-a x} \sin p x d x=\frac{p}{a^{2}+p^{2}}$

[Any standard results that you use should be stated precisely.]

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• # A1.3

(i) Let $T: H_{1} \rightarrow H_{2}$ be a continuous linear map between two Hilbert spaces $H_{1}, H_{2}$. Define the adjoint $T^{*}$ of $T$. Explain what it means to say that $T$ is Hermitian or unitary.

Let $\phi: \mathbb{R} \rightarrow \mathbb{C}$ be a bounded continuous function. Show that the map

$T: L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{R})$

with $T f(x)=\phi(x) f(x+1)$ is a continuous linear map and find its adjoint. When is $T$ Hermitian? When is it unitary?

(ii) Let $C$ be a closed, non-empty, convex subset of a real Hilbert space $H$. Show that there exists a unique point $x_{o} \in C$ with minimal norm. Show that $x_{o}$ is characterised by the property

$\left\langle x_{o}-x, x_{o}\right\rangle \leqslant 0 \quad \text { for all } x \in C .$

Does this result still hold when $C$ is not closed or when $C$ is not convex? Justify your answers.

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• # A2.3 B2.2

(i) Define the dual of a normed vector space $(E,\|\cdot\|)$. Show that the dual is always a complete normed space.

Prove that the vector space $\ell_{1}$, consisting of those real sequences $\left(x_{n}\right)_{n=1}^{\infty}$ for which the norm

$\left\|\left(x_{n}\right)\right\|_{1}=\sum_{n=1}^{\infty}\left|x_{n}\right|$

is finite, has the vector space $\ell_{\infty}$ of all bounded sequences as its dual.

(ii) State the Stone-Weierstrass approximation theorem.

Let $K$ be a compact subset of $\mathbb{R}^{n}$. Show that every $f \in C_{\mathbb{R}}(K)$ can be uniformly approximated by a sequence of polynomials in $n$ variables.

Let $f$ be a continuous function on $[0,1] \times[0,1]$. Deduce that

$\int_{0}^{1}\left(\int_{0}^{1} f(x, y) d x\right) d y=\int_{0}^{1}\left(\int_{0}^{1} f(x, y) d y\right) d x$

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• # A3.3 B3.2

(i) Let $p$ be a point of the compact interval $I=[a, b] \subset \mathbb{R}$ and let $\delta_{p}: C(I) \rightarrow \mathbb{R}$ be defined by $\delta_{p}(f)=f(p)$. Show that

$\delta_{p}:\left(C(I),\|\cdot\|_{\infty}\right) \rightarrow \mathbb{R}$

is a continuous, linear map but that

$\delta_{p}:\left(C(I),\|\cdot\|_{1}\right) \rightarrow \mathbb{R}$

is not continuous.

(ii) Consider the space $C^{(n)}(I)$ of $n$-times continuously differentiable functions on the interval $I$. Write

$\|f\|_{\infty}^{(n)}=\sum_{k=0}^{n}\left\|f^{(k)}\right\|_{\infty} \quad \text { and } \quad\|f\|_{1}^{(n)}=\sum_{r=0}^{n}\left\|f^{(k)}\right\|_{1}$

for $f \in C^{(n)}(I)$. Show that $\left(C^{(n)}(I),\left\|^{\prime} \cdot\right\|_{\infty}^{(n)}\right)$ is a complete normed space. Is the space $\left(C^{(n)}(I),\|\cdot\|_{1}^{(n)}\right)$ also complete?

Let $f: I \rightarrow I$ be an $n$-times continuously differentiable map and define

$\mu_{f}: C^{(n)}(I) \rightarrow C^{(n)}(I) \quad \text { by } \quad g \mapsto g \circ f .$

Show that $\mu_{f}$ is a continuous linear map when $C^{(n)}(I)$ is equipped with the norm $\|\cdot\|_{\infty}^{(n)}$.

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• # A4.3

(i) State the Monotone Convergence Theorem and explain briefly how to prove it.

(ii) For which real values of $\alpha$ is $x^{-\alpha} \log x \in L^{1}((1, \infty))$ ?

Let $p>0$. Using the Monotone Convergence Theorem and the identity

$\frac{1}{x^{p}(x-1)}=\sum_{n=0}^{\infty} \frac{1}{x^{p+n+1}}$

prove carefully that

$\int_{1}^{\infty} \frac{\log x}{x^{p}(x-1)} d x=\sum_{n=0}^{\infty} \frac{1}{(n+p)^{2}}$

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• # A1.3

(i) Let $P_{r}\left(e^{i \theta}\right)$ be the real part of $\frac{1+r e^{i \theta}}{1-r e^{i \theta}}$. Establish the following properties of $P_{r}$ for $0 \leqslant r<1$ : (a) $0; (b) $P_{r}\left(e^{i \theta}\right) \leqslant P_{r}\left(e^{i \delta}\right)$ for $0<\delta \leqslant|\theta| \leqslant \pi$; (c) $P_{r}\left(e^{i \theta}\right) \rightarrow 0$, uniformly on $0<\delta \leqslant|\theta| \leqslant \pi$, as $r$ increases to 1 .

(ii) Suppose that $f \in L^{1}(\mathbf{T})$, where $\mathbf{T}$ is the unit circle $\left\{e^{i \theta}:-\pi \leqslant \theta \leqslant \pi\right\}$. By definition, $\|f\|_{1}=\frac{1}{2 \pi} \int_{-\pi}^{\pi}\left|f\left(e^{i \theta}\right)\right| d \theta$. Let

$P_{r}(f)\left(e^{i \theta}\right)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} P_{r}\left(e^{i(\theta-t)}\right) f\left(e^{i t}\right) d t$

Show that $P_{r}(f)$ is a continuous function on $\mathbf{T}$, and that $\left\|P_{r}(f)\right\|_{1} \leqslant\|f\|_{1}$.

[You may assume without proof that $\frac{1}{2 \pi} \int_{-\pi}^{\pi} P_{r}\left(e^{i \theta}\right) d \theta=1$.]

Show that $P_{r}(f) \rightarrow f$, uniformly on $\mathbf{T}$ as $r$ increases to 1 , if and only if $f$ is a continuous function on $\mathbf{T}$.

Show that $\left\|P_{r}(f)-f\right\|_{1} \rightarrow 0$ as $r$ increases to 1 .

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• # A2.3 B2.2

(i) State and prove the parallelogram law for Hilbert spaces.

Suppose that $K$ is a closed linear subspace of a Hilbert space $H$ and that $x \in H$. Show that $x$ is orthogonal to $K$ if and only if 0 is the nearest point to $x$ in $K$.

(ii) Suppose that $H$ is a Hilbert space and that $\phi$ is a continuous linear functional on $H$ with $\|\phi\|=1$. Show that there is a sequence $\left(h_{n}\right)$ of unit vectors in $H$ with $\phi\left(h_{n}\right)$ real and $\phi\left(h_{n}\right)>1-1 / n$.

Show that $h_{n}$ converges to a unit vector $h$, and that $\phi(h)=1$.

Show that $h$ is orthogonal to $N$, the null space of $\phi$, and also that $H=N \oplus \operatorname{span}(h)$.

Show that $\phi(k)=\langle k, h\rangle$, for all $k \in H$.

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• # A3.3 B3.2

(i) Suppose that $\left(f_{n}\right)$ is a decreasing sequence of continuous real-valued functions on a compact metric space $(X, d)$ which converges pointwise to 0 . By considering sets of the form $B_{n}=\left\{x: f_{n}(x)<\epsilon\right\}$, for $\epsilon>0$, or otherwise, show that $f_{n}$ converges uniformly to 0 .

Can the condition that $\left(f_{n}\right)$ is decreasing be dropped? Can the condition that $(X, d)$ is compact be dropped? Justify your answers.

(ii) Suppose that $k$ is a positive integer. Define polynomials $p_{n}$ recursively by

$p_{0}=0, \quad p_{n+1}(t)=p_{n}(t)+\left(t-p_{n}^{k}(t)\right) / k$

Show that $0 \leqslant p_{n}(t) \leqslant p_{n+1}(t) \leqslant t^{1 / k}$, for $t \in[0,1]$, and show that $p_{n}(t)$ converges to $t^{1 / k}$ uniformly on $[0,1]$.

[You may wish to use the identity $a^{k}-b^{k}=(a-b)\left(a^{k-1}+a^{k-2} b+\ldots+b^{k-1}\right)$.]

Suppose that $A$ is a closed subalgebra of the algebra $C(X)$ of continuous real-valued functions on a compact metric space $(X, d)$, equipped with the uniform norm, and suppose that $A$ has the property that for each $x \in X$ there exists $a \in A$ with $a(x) \neq 0$. Show that there exists $h \in A$ such that $0 for all $x \in X$.

Show that $h^{1 / k} \in A$ for each positive integer $k$, and show that $A$ contains the constant functions.

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• # A4.3

Define the distribution function $\Phi_{f}$ of a non-negative measurable function $f$ on the interval $I=[0,1]$. Show that $\Phi_{f}$ is a decreasing non-negative function on $[0, \infty]$ which is continuous on the right.

Define the Lebesgue integral $\int_{I} f d m$. Show that $\int_{I} f d m=0$ if and only if $f=0$ almost everywhere.

Suppose that $f$ is a non-negative Riemann integrable function on $[0,1]$. Show that there are an increasing sequence $\left(g_{n}\right)$ and a decreasing sequence $\left(h_{n}\right)$ of non-negative step functions with $g_{n} \leqslant f \leqslant h_{n}$ such that $\int_{0}^{1}\left(h_{n}(x)-g_{n}(x)\right) d x \rightarrow 0$.

Show that the functions $g=\lim _{n} g_{n}$ and $h=\lim _{n} h_{n}$ are equal almost everywhere, that $f$ is measurable and that the Lebesgue integral $\int_{I} f d m$ is equal to the Riemann integral $\int_{0}^{1} f(x) d x$.

Suppose that $j$ is a Riemann integrable function on $[0,1]$ and that $j(x)>0$ for all $x$. Show that $\int_{0}^{1} j(x) d x>0$.

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• # A1 $. 3 \quad$

(i) Define the adjoint of a bounded, linear map $u: H \rightarrow H$ on the Hilbert space $H$. Find the adjoint of the map

$u: H \rightarrow H ; \quad x \mapsto \phi(x) a$

where $a, b \in H$ and $\phi \in H^{*}$ is the linear map $x \mapsto\langle b, x\rangle$.

Now let $J$ be an incomplete inner product space and $u: J \rightarrow J$ a bounded, linear map. Is it always true that there is an adjoint $u^{*}: J \rightarrow J$ ?

(ii) Let $\mathcal{H}$ be the space of analytic functions $f: \mathbb{D} \rightarrow \mathbb{C}$ on the unit disc $\mathbb{D}$ for which

$\iint_{\mathbb{D}}|f(z)|^{2} d x d y<\infty \quad(z=x+i y)$

You may assume that this is a Hilbert space for the inner product:

$\langle f, g\rangle=\iint_{\mathbb{D}} \overline{f(z)} g(z) d x d y .$

Show that the functions $u_{k}: z \mapsto \alpha_{k} z^{k}(k=0,1,2, \ldots)$ form an orthonormal sequence in $\mathcal{H}$ when the constants $\alpha_{k}$ are chosen appropriately.

Prove carefully that every function $f \in \mathcal{H}$ can be written as the sum of a convergent series $\sum_{k=0}^{\infty} f_{k} u_{k}$ in $\mathcal{H}$ with $f_{k} \in \mathbb{C}$.

For each smooth curve $\gamma$ in the disc $\mathbb{D}$ starting from 0 , prove that

$\phi: \mathcal{H} \rightarrow \mathbb{C} ; \quad f \mapsto \int_{\gamma} f(z) d z$

is a continuous, linear map. Show that the norm of $\phi$ satisfies

$\|\phi\|^{2}=\frac{1}{\pi} \log \left(\frac{1}{1-|w|^{2}}\right)$

where $w$ is the endpoint of $\gamma$.

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• # A2.3 B2.2

(i) State the Stone-Weierstrass theorem for complex-valued functions. Use it to show that the trigonometric polynomials are dense in the space $C(\mathbb{T})$ of continuous, complexvalued functions on the unit circle $\mathbb{T}$ with the uniform norm.

Show further that, for $f \in C(\mathbb{T})$, the $n$th Fourier coefficient

$\widehat{f}(n)=\frac{1}{2 \pi} \int_{0}^{2 \pi} f\left(e^{i \theta}\right) e^{-i n \theta} d \theta$

tends to 0 as $|n|$ tends to infinity.

(ii) (a) Let $X$ be a normed space with the property that the series $\sum_{n=1}^{\infty} x_{n}$ converges whenever $\left(x_{n}\right)$ is a sequence in $X$ with $\sum_{n=1}^{\infty}\left\|x_{n}\right\|$ convergent. Show that $X$ is a Banach space.

(b) Let $K$ be a compact metric space and $L$ a closed subset of $K$. Let $R: C(K) \rightarrow$ $C(L)$ be the map sending $f \in C(K)$ to its restriction $R(f)=f \mid L$ to $L$. Show that $R$ is a bounded, linear map and that its image is a subalgebra of $C(L)$ separating the points of

Show further that, for each function $g$ in the image of $R$, there is a function $f \in C(K)$ with $R(f)=g$ and $\|f\|_{\infty}=\|g\|_{\infty}$. Deduce that every continuous, complexvalued function on $L$ can be extended to a continuous function on all of $K$.

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• # A3.3 B3.2

(i) Define the notion of a measurable function between measurable spaces. Show that a continuous function $\mathbb{R}^{2} \rightarrow \mathbb{R}$ is measurable with respect to the Borel $\sigma$-fields on $\mathbb{R}^{2}$ and $\mathbb{R}$.

By using this, or otherwise, show that, when $f, g: X \rightarrow \mathbb{R}$ are measurable with respect to some $\sigma$-field $\mathcal{F}$ on $X$ and the Borel $\sigma$-field on $\mathbb{R}$, then $f+g$ is also measurable.

(ii) State the Monotone Convergence Theorem for $[0, \infty]$-valued functions. Prove the Dominated Convergence Theorem.

[You may assume the Monotone Convergence Theorem but any other results about integration that you use will need to be stated carefully and proved.]

Let $X$ be the real Banach space of continuous real-valued functions on $[0,1]$ with the uniform norm. Fix $u \in X$ and define

$T: X \rightarrow \mathbb{R} ; \quad f \mapsto \int_{0}^{1} f(t) u(t) d t$

Show that $T$ is a bounded, linear map with norm

$\|T\|=\int_{0}^{1}|u(t)| d t$

Is it true, for every choice of $u$, that there is function $f \in X$ with $\|f\|=1$ and $\|T(f)\|=\|T\|$ ?

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• # A4.3

Write an account of the classical sequence spaces: $\ell_{p}(1 \leqslant p \leqslant \infty)$ and $c_{0}$. You should define them, prove that they are Banach spaces, and discuss their properties, including their dual spaces. Show that $\ell_{\infty}$ is inseparable but that $c_{0}$ and $\ell_{p}$ for $1 \leqslant p<\infty$ are separable.

Prove that, if $T: X \rightarrow Y$ is an isomorphism between two Banach spaces, then

$T^{*}: Y^{*} \rightarrow X^{*} ; \quad f \mapsto f \circ T$

is an isomorphism between their duals.

Hence, or otherwise, show that no two of the spaces $c_{0}, \ell_{1}, \ell_{2}, \ell_{\infty}$ are isomorphic.

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