Linear Analysis

• 1.II.22F

  Linear Analysis

  Part II, 2005

  Let $K$ be a compact Hausdorff space, and let $C(K)$ denote the Banach space of continuous, complex-valued functions on $K$, with the supremum norm. Define what it means for a set $S \subset C(K)$ to be totally bounded, uniformly bounded, and equicontinuous.

  Show that $S$ is totally bounded if and only if it is both uniformly bounded and equicontinuous.

  Give, with justification, an example of a Banach space $X$ and a subset $S \subset X$ such that $S$ is bounded but not totally bounded.

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• 2.II.22F

  Linear Analysis

  Part II, 2005

  Let $X$ and $Y$ be Banach spaces. Define what it means for a linear operator $T: X \rightarrow Y$ to be compact. For a linear operator $T: X \rightarrow X$, define the spectrum, point spectrum, and resolvent set of $T$.

  Now let $H$ be a complex Hilbert space. Define what it means for a linear operator $T: H \rightarrow H$ to be self-adjoint. Suppose $e_{1}, e_{2}, \ldots$ is an orthonormal basis for $H$. Define a linear operator $T: H \rightarrow H$ by setting $T e_{i}=\frac{1}{i} e_{i}$. Is $T$ compact? Is $T$ self-adjoint? Justify your answers. Describe, with proof, the spectrum, point spectrum, and resolvent set of $T$.

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• 3.II.21F

  Linear Analysis

  Part II, 2005

  Let $X$ be a normed vector space. Define the dual $X^{*}$ of $X$. Define the normed vector spaces $l^{s}=l^{s}(\mathbb{C})$ for all $1 \leqslant s \leqslant \infty$. [You are not required to prove that the norms you have given are indeed norms.]

  Now let $1<p, q<\infty$ be such that $p^{-1}+q^{-1}=1$. Show that $\left(l^{q}\right)^{*}$ is isometrically isomorphic to $l^{p}$ as a normed vector space. [You may assume any standard inequalities.]

  Show by a similar argument that $\left(l^{1}\right)^{*}$ is isomorphic to $l^{\infty}$. Does your argument also show that $\left(l^{\infty}\right)^{*}$ is isomorphic to $l^{1}$ ? If not, where does it fail?

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• 4.II.22F

  Linear Analysis

  Part II, 2005

  Let $X$ and $Y$ be normed vector spaces. Show that a linear map $T: X \rightarrow Y$ is continuous if and only if it is bounded.

  Now let $X, Y, Z$ be Banach spaces. We say that a map $F: X \times Y \rightarrow Z$ is bilinear

  $$\begin{aligned} &F(\alpha x+\beta y, z)=\alpha F(x, z)+\beta F(y, z), \text { for all scalars } \alpha, \beta \text { and } x, y \in X, z \in Y \\ &F(x, \alpha y+\beta z)=\alpha F(x, y)+\beta F(x, z), \text { for all scalars } \alpha, \beta \text { and } x \in X, y, z \in Y . \end{aligned}$$

  Suppose that $F$ is bilinear and is continuous in each variable separately. Show that there exists a constant $M \geqslant 0$ such that

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

  for all $x \in X, y \in Y$.

  [Hint: For each fixed $x \in X$, consider the map $y \mapsto F(x, y)$ from $Y$ to $Z$.]

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• 1.II.22G

  Linear Analysis

  Part II, 2006

  Let $U$ be a vector space. Define what it means for two norms $\|\cdot\|_{1}$ and $\left\|_{\cdot} \cdot\right\|_{2}$ on $U$ to be Lipschitz equivalent. Give an example of a vector space and two norms which are not Lipschitz equivalent.

  Show that, if $U$ is finite dimensional, all norms on $U$ are Lipschitz equivalent. Deduce that a finite dimensional subspace of a normed vector space is closed.

  Show that a normed vector space $W$ is finite dimensional if and only if $W$ contains a non-empty open set with compact closure.

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• 2.II.22G

  Linear Analysis

  Part II, 2006

  Let $X$ be a metric space. Define what it means for a subset $E \subset X$ to be of first or second category. State and prove a version of the Baire category theorem. For $1 \leqslant p \leqslant \infty$, show that the set $\ell_{p}$ is of first category in the normed space $\ell_{r}$ when $r>p$ and $\ell_{r}$ is given its standard norm. What about $r=p$ ?

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• 3.II.21G

  Linear Analysis

  Part II, 2006

  Let $X$ be a complex Banach space. We say a sequence $x^{i} \in X$ converges to $x \in X$ weakly if $\phi\left(x^{i}\right) \rightarrow \phi(x)$ for all $\phi \in X^{*}$. Let $T: X \rightarrow Y$ be bounded and linear. Show that if $x^{i}$ converges to $x$ weakly, then $T x^{i}$ converges to $T x$ weakly.

  Now let $X=\ell_{2}$. Show that for a sequence $x^{i} \in X, i=1,2, \ldots$, with $\left\|x^{i}\right\| \leqslant 1$, there exists a subsequence $x^{i_{k}}$ such that $x^{i_{k}}$ converges weakly to some $x \in X$ with $\|x\| \leqslant 1$.

  Now let $Y=\ell_{1}$, and show that $y^{i} \in Y$ converges to $y \in Y$ weakly if and only if $y^{i} \rightarrow y$ in the usual sense.

  Define what it means for a linear operator $T: X \rightarrow Y$ to be compact, and deduce from the above that any bounded linear $T: \ell_{2} \rightarrow \ell_{1}$ is compact.

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• 4.II.22G

  Linear Analysis

  Part II, 2006

  Let $H$ be a complex Hilbert space. Define what it means for a linear operator $T: H \rightarrow H$ to be self-adjoint. State a version of the spectral theorem for compact selfadjoint operators on a Hilbert space. Give an example of a Hilbert space $H$ and a compact self-adjoint operator on $H$ with infinite dimensional range. Define the notions spectrum, point spectrum, and resolvent set, and describe these in the case of the operator you wrote down. Justify your answers.

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• 1.II.22G

  Linear Analysis

  Part II, 2007

  Let $X$ be a normed vector space over $\mathbb{R}$. Define the dual space $X^{*}$ and show directly that $X^{*}$ is a Banach space. Show that the map $\phi: X \rightarrow X^{* *}$ defined by $\phi(x) v=v(x)$, for all $x \in X, v \in X^{*}$, is a linear map. Using the Hahn-Banach theorem, show that $\phi$ is injective and $|\phi(x)|=|x|$.

  Give an example of a Banach space $X$ for which $\phi$ is not surjective. Justify your answer.

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• 2.II.22G

  Linear Analysis

  Part II, 2007

  Let $X$ be a Banach space, $Y$ a normed vector space, and $T: X \rightarrow Y$ a bounded linear map. Assume that $T(X)$ is of second category in $Y$. Show that $T$ is surjective and $T(\mathcal{U})$ is open whenever $\mathcal{U}$ is open. Show that, if $T$ is also injective, then $T^{-1}$ exists and is bounded.

  Give an example of a continuous map $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(\mathbb{R})$ is of second category in $\mathbb{R}$ but $f$ is not surjective. Give an example of a continuous surjective map $f: \mathbb{R} \rightarrow \mathbb{R}$ which does not take open sets to open sets.

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• 3.II.21G

  Linear Analysis

  Part II, 2007

  State and prove the Arzela-Ascoli theorem.

  Let $N$ be a positive integer. Consider the subset $\mathcal{S}_{N} \subset C([0,1])$ consisting of all thrice differentiable solutions to the differential equation

  $f^{\prime \prime}=f+\left(f^{\prime}\right)^{2} \quad$ with $\quad|f(0)| \leqslant N, \quad|f(1)| \leqslant N, \quad\left|f^{\prime}(0)\right| \leqslant N, \quad\left|f^{\prime}(1)\right| \leqslant N .$

  Show that this set is totally bounded as a subset of $C([0,1])$.

  [It may be helpful to consider interior maxima.]

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• 4.II.22G

  Linear Analysis

  Part II, 2007

  Let $X$ be a Banach space and $T: X \rightarrow X$ a bounded linear map. Define the spectrum $\sigma(T)$, point spectrum $\sigma_{p}(T)$, resolvent $R_{T}(\lambda)$, and resolvent set $\rho(T)$. Show that the spectrum is a closed and bounded subset of $\mathbb{C}$. Is the point spectrum always closed? Justify your answer.

  Now suppose $H$ is a Hilbert space, and $T: H \rightarrow H$ is self-adjoint. Show that the point spectrum $\sigma_{p}(T)$ is real.

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• 1.II.22F

  Linear Analysis

  Part II, 2008

  Suppose $p$ and $q$ are real numbers with $p^{-1}+q^{-1}=1$ and $p, q>1$. Show, quoting any results on convexity that you need, that

  $$a^{1 / p} b^{1 / q} \leqslant \frac{a}{p}+\frac{b}{q}$$

  for all real positive $a$ and $b$.

  Define the space $l^{p}$ and show that it is a complete normed vector space.

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• 2.II.22F

  Linear Analysis

  Part II, 2008

  State and prove the principle of uniform boundedness.

  [You may assume the Baire category theorem.]

  Suppose that $X, Y$ and $Z$ are Banach spaces. Suppose that

  $$F: X \times Y \rightarrow Z$$

  is linear and continuous in each variable separately, that is to say that, if $y$ is fixed,

  $$F(\cdot, y): X \rightarrow Z$$

  is a continuous linear map and, if $x$ is fixed,

  $$F(x, \cdot): Y \rightarrow Z$$

  is a continuous linear map. Show that there exists an $M$ such that

  $$\|F(x, y)\|_{Z} \leqslant M\|x\|_{X}\|y\|_{Y}$$

  for all $x \in X, y \in Y$. Deduce that $F$ is continuous.

  Suppose $X, Y, Z$ and $W$ are Banach spaces. Suppose that

  $$G: X \times Y \times W \rightarrow Z$$

  is linear and continuous in each variable separately. Does it follow that $G$ is continuous? Give reasons.

  Suppose that $X, Y$ and $Z$ are Banach spaces. Suppose that

  $$H: X \times Y \rightarrow Z$$

  is continuous in each variable separately. Does it follow that $H$ is continuous? Give reasons.

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• 3.II.21F

  Linear Analysis

  Part II, 2008

  State and prove the Stone-Weierstrass theorem for real-valued functions. You may assume that the function $x \mapsto|x|$ can be uniformly approximated by polynomials on any interval $[-k, k]$.

  Suppose that $0<a<b<1$. Let $\mathcal{F}$ be the set of functions which can be uniformly approximated on $[a, b]$ by polynomials with integer coefficients. By making appropriate use of the identity

  $$\frac{1}{2}=\frac{x}{1-(1-2 x)}=\sum_{n=0}^{\infty} x(1-2 x)^{n}$$

  or otherwise, show that $\mathcal{F}=\mathcal{C}([a, b])$.

  Is it true that every continuous function on $[0, b]$ can be uniformly approximated by polynomials with integer coefficients?

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• 4.II.22F

  Linear Analysis

  Part II, 2008

  Let $H$ be a Hilbert space. Show that if $V$ is a closed subspace of $H$ then any $f \in H$ can be written as $f=v+w$ with $v \in V$ and $w \perp V$.

  Suppose $U: H \rightarrow H$ is unitary (that is to say $U U^{*}=U^{*} U=I$ ). Let

  $$A_{n} f=\frac{1}{n} \sum_{k=0}^{n-1} U^{k} f$$

  and consider

  $$X=\{g-U g: g \in H\}$$

  (i) Show that $U$ is an isometry and $\left\|A_{n}\right\| \leqslant 1$.

  (ii) Show that $X$ is a subspace of $H$ and $A_{n} f \rightarrow 0$ as $n \rightarrow \infty$ whenever $f \in X$.

  (iii) Let $V$ be the closure of $X$. Show that $A_{n} v \rightarrow 0$ as $n \rightarrow \infty$ whenever $v \in V$.

  (iv) Show that, if $w \perp X$, then $U w=w$. Deduce that, if $w \perp V$, then $U w=w$.

  (v) If $f \in H$ show that there is a $w \in H$ such that $A_{n} f \rightarrow w$ as $n \rightarrow \infty$.

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• Paper 3, Section II, H

  Linear Analysis

  Part II, 2009

  (a) State the Arzela-Ascoli theorem, explaining the meaning of all concepts involved.

  (b) Prove the Arzela-Ascoli theorem.

  (c) Let $K$ be a compact topological space. Let $\left(f_{n}\right)_{n \in \mathbb{N}}$ be a sequence in the Banach space $C(K)$ of real-valued continuous functions over $K$ equipped with the supremum norm $\|\cdot\|$. Assume that for every $x \in K$, the sequence $f_{n}(x)$ is monotone increasing and that $f_{n}(x) \rightarrow f(x)$ for some $f \in C(K)$. Show that $\left\|f_{n}-f\right\| \rightarrow 0$ as $n \rightarrow \infty$.

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• Paper 2, Section II, H

  Linear Analysis

  Part II, 2009

  For $1 \leqslant p<\infty$ and a sequence $x=\left(x_{1}, x_{2}, \ldots\right)$, where $x_{j} \in \mathbb{C}$ for all $j \geqslant 1$, let $\|x\|_{p}=\left(\sum_{j=1}^{\infty}\left|x_{j}\right|^{p}\right)^{1 / p} .$

  Let $\ell^{p}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C}\right.$ for all $j \geqslant 1$ and $\left.\|x\|_{p}<\infty\right\}$.

  (a) Let $p, q>1$ with $1 / p+1 / q=1, x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{p}$ and $y=\left(y_{1}, y_{2}, \ldots\right) \in \ell^{q}$. Prove Hölder's inequality:

  $$\sum_{j=1}^{\infty}\left|x_{j}\left\|y_{j} \mid \leqslant\right\| x\left\|_{p}\right\| y \|_{q}\right.$$

  (b) Use Hölder's inequality to prove the triangle inequality (known, in this case, as the Minkowski inequality):

  $$\|x+y\|_{p} \leqslant\|x\|_{p}+\|y\|_{p} \quad \text { for every } x, y \in \ell^{p} \quad \text { and every } 1<p<\infty$$

  (c) Let $2 \leqslant p<\infty$ and let $K$ be a closed, convex subset of $\ell^{p}$. Let $x \in \ell^{p}$ with $x \notin K$. Prove that there exists $y \in K$ such that

  $$\|x-y\|=\inf _{z \in K}\|x-z\| .$$

  [You may use without proof the fact that for every $2 \leqslant p<\infty$ and for every $x, y \in \ell^{p}$,

  $$\left.\|x+y\|_{p}^{p}+\|x-y\|_{p}^{p} \leqslant 2^{p-1}\left(\|x\|_{p}^{p}+\|y\|_{p}^{p}\right) .\right]$$

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• Paper 4, Section II, H

  Linear Analysis

  Part II, 2009

  Let $X$ be a Banach space and let $T: X \rightarrow X$ be a bounded linear map.

  (a) Define the spectrum $\sigma(T)$, the resolvent set $\rho(T)$ and the point spectrum $\sigma_{p}(T)$ of $T$.

  (b) What does it mean for $T$ to be a compact operator?

  (c) Show that if $T$ is a compact operator on $X$ and $a>0$, then $T$ has at most finitely many linearly independent eigenvectors with eigenvalues having modulus larger than $a$.

  [You may use without proof the fact that for any finite dimensional proper subspace $Y$ of a Banach space $Z$, there exists $x \in Z$ with $\|x\|=1$ and $\operatorname{dist}(x, Y)=\inf _{y \in Y}\|x-y\|=1$.]

  (d) For a sequence $\left(\lambda_{n}\right)_{n \geqslant 1}$ of complex numbers, let $T: \ell^{2} \rightarrow \ell^{2}$ be defined by

  $$T\left(x_{1}, x_{2}, \ldots\right)=\left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots\right) .$$

  Give necessary and sufficient conditions on the sequence $\left(\lambda_{n}\right)_{n} \geqslant 1$ for $T$ to be compact, and prove your assertion.

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• Paper 1, Section II, H

  Linear Analysis

  Part II, 2009

  (a) State and prove the Baire category theorem.

  (b) Let $X$ be a normed space. Show that every proper linear subspace $V \subset X$ has empty interior.

  (c) Let $\mathcal{P}$ be the vector space of all real polynomials in one variable. Using the Baire category theorem and the result from (b), prove that for any norm $\|\cdot\|$ on $\mathcal{P}$, the normed space $(\mathcal{P},\|\cdot\|)$ is not a Banach space.

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• Paper 1, Section II, H

  Linear Analysis

  Part II, 2010

  a) State and prove the Banach-Steinhaus Theorem.

  [You may use the Baire Category Theorem without proving it.]

  b) Let $X$ be a (complex) normed space and $S \subset X$. Prove that if $\{f(x): x \in S\}$ is a bounded set in $\mathbb{C}$ for every linear functional $f \in X^{*}$ then there exists $K \geqslant 0$ such that $\|x\| \leqslant K$ for all $x \in S .$

  [You may use here the following consequence of the Hahn-Banach Theorem without proving it: for a given $x \in X$, there exists $f \in X^{*}$ with $\|f\|=1$ and $|f(x)|=\|x\|$.]

  c) Conclude that if two norms $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ on a (complex) vector space $V$ are not equivalent, there exists a linear functional $f: V \rightarrow \mathbb{C}$ which is continuous with respect to one of the two norms, and discontinuous with respect to the other.

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• Paper 2, Section II, H

  Linear Analysis

  Part II, 2010

  For a sequence $x=\left(x_{1}, x_{2}, \ldots\right)$ with $x_{j} \in \mathbb{C}$ for all $j \geqslant 1$, let

  $$\|x\|_{\infty}:=\sup _{j \geqslant 1}\left|x_{j}\right|$$

  and $\ell^{\infty}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C}\right.$ for all $j \geqslant 1$ and $\left.\|x\|_{\infty}<\infty\right\}$.

  a) Prove that $\ell^{\infty}$ is a Banach space.

  b) Define

  $$c_{0}=\left\{x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{\infty}: \lim _{j \rightarrow \infty} x_{j}=0\right\}$$

  and

  $$\ell^{1}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C} \text { for all } j \in \mathbb{N} \text { and }\|x\|_{1}=\sum_{\ell=1}^{\infty}\left|x_{\ell}\right|<\infty\right\}$$

  Show that $c_{0}$ is a closed subspace of $\ell^{\infty}$. Show that $c_{0}^{*} \simeq \ell^{1}$.

  [Hint: find an isometric isomorphism from $\ell^{1}$ to $\left.c_{0}^{*} .\right]$

  c) Let

  $$c_{00}=\left\{x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{\infty}: x_{j}=0 \text { for all } j \text { large enough }\right\} .$$

  Is $c_{00}$ a closed subspace of $\ell^{\infty} ?$ If not, what is the closure of $c_{00} ?$

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• Paper 3, Section II, H

  Linear Analysis

  Part II, 2010

  State and prove the Stone-Weierstrass theorem for real-valued functions.

  [You may use without proof the fact that the function $s \rightarrow|s|$ can be uniformly approximated by polynomials on $[-1,1] .]$

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• Paper 4, Section II, H

  Linear Analysis

  Part II, 2010

  Let $X$ be a Banach space.

  a) What does it mean for a bounded linear map $T: X \rightarrow X$ to be compact?

  b) Let $\mathcal{B}(X)$ be the Banach space of all bounded linear maps $S: X \rightarrow X$. Let $\mathcal{B}_{0}(X)$ be the subset of $\mathcal{B}(X)$ consisting of all compact operators. Show that $\mathcal{B}_{0}(X)$ is a closed subspace of $\mathcal{B}(X)$. Show that, if $S \in \mathcal{B}(X)$ and $T \in \mathcal{B}_{0}(X)$, then $S T, T S \in \mathcal{B}_{0}(X)$.

  c) Let

  $$X=\ell^{2}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C} \quad \text { and } \quad\|x\|_{2}^{2}=\sum_{j=1}^{\infty}\left|x_{j}\right|^{2}<\infty\right\}$$

  and $T: X \rightarrow X$ be defined by

  $$(T x)_{k}=\frac{x_{k+1}}{k+1} .$$

  Is $T$ compact? What is the spectrum of $T ?$ Explain your answers.

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• Paper 1, Section II, G

  Linear Analysis

  Part II, 2011

  State a version of the Stone-Weierstrass Theorem for real-valued functions on a compact metric space.

  Suppose that $K:[0,1]^{2} \rightarrow \mathbb{R}$ is a continuous function. Show that $K(x, y)$ may be uniformly approximated by functions of the form $\sum_{i=1}^{n} f_{i}(x) g_{i}(y)$ with $f_{i}, g_{i}:[0,1] \rightarrow \mathbb{R}$ continuous.

  Let $X, Y$ be Banach spaces and suppose that $T: X \rightarrow Y$ is a bounded linear operator. What does it mean to say that $T$ is finite-rank? What does it mean to say that $T$ is compact? Give an example of a bounded linear operator from $C[0,1]$ to itself which is not compact.

  Suppose that $\left(T_{n}\right)_{n=1}^{\infty}$ is a sequence of finite-rank operators and that $T_{n} \rightarrow T$ in the operator norm. Briefly explain why the $T_{n}$ are compact. Show that $T$ is compact.

  Hence, show that the integral operator $T: C[0,1] \rightarrow C[0,1]$ defined by

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

  is compact.

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• Paper 2, Section II, G

  Linear Analysis

  Part II, 2011

  State and prove the Baire Category Theorem. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. For $x \in \mathbb{R}$, define

  $$\omega_{f}(x)=\inf _{\delta>0} \sup _{\substack{|y-x| \leqslant \delta \\\left|y^{\prime}-x\right| \leqslant \delta}}\left|f(y)-f\left(y^{\prime}\right)\right|$$

  Show that $f$ is continuous at $x$ if and only if $\omega_{f}(x)=0$.

  Show that for any $\epsilon>0$ the set $\left\{x \in \mathbb{R}: \omega_{f}(x)<\epsilon\right\}$ is open.

  Hence show that the set of points at which $f$ is continuous cannot be precisely the set $\mathbb{Q}$ of rationals.

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• Paper 3, Section II, G

  Linear Analysis

  Part II, 2011

  Let $H$ be a complex Hilbert space with orthonormal basis $\left(e_{n}\right)_{n=-\infty}^{\infty}$ Let $T: H \rightarrow H$ be a bounded linear operator. What is meant by the spectrum $\sigma(T)$ of $T$ ?

  Define $T$ by setting $T\left(e_{n}\right)=e_{n-1}+e_{n+1}$ for $n \in \mathbb{Z}$. Show that $T$ has a unique extension to a bounded, self-adjoint linear operator on $H$. Determine the norm $\|T\|$. Exhibit, with proof, an element of $\sigma(T)$.

  Show that $T$ has no eigenvectors. Is $T$ compact?

  [General results from spectral theory may be used without proof. You may also use the fact that if a sequence $\left(x_{n}\right)$ satisfies a linear recurrence $\lambda x_{n}=x_{n-1}+x_{n+1}$ with $\lambda \in \mathbb{R}$, $|\lambda| \leqslant 2, \lambda \neq 0$, then it has the form $x_{n}=A \alpha^{n} \sin \left(\theta_{1} n+\theta_{2}\right)$ or $x_{n}=(A+n B) \alpha^{n}$, where $A, B, \alpha \in \mathbb{R}$ and $0 \leqslant \theta_{1}<\pi,\left|\theta_{2}\right| \leqslant \pi / 2$.]

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• Paper 4, Section II, G

  Linear Analysis

  Part II, 2011

  State Urysohn's Lemma. State and prove the Tietze Extension Theorem.

  Let $X, Y$ be two topological spaces. We say that the extension property holds if, whenever $S \subseteq X$ is a closed subset and $f: S \rightarrow Y$ is a continuous map, there is a continuous function $\tilde{f}: X \rightarrow Y$ with $\left.\tilde{f}\right|_{S}=f$.

  For each of the following three statements, say whether it is true or false. Briefly justify your answers.

  1. If $X$ is a metric space and $Y=[-1,1]$ then the extension property holds.

  2. If $X$ is a compact Hausdorff space and $Y=\mathbb{R}$ then the extension property holds.

  3. If $X$ is an arbitrary topological space and $Y=[-1,1]$ then the extension property holds.

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• Paper 3, Section II, G

  Linear Analysis

  Part II, 2012

  State the closed graph theorem.

  (i) Let $X$ be a Banach space and $Y$ a vector space. Suppose that $Y$ is endowed with two norms $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ and that there is a constant $c>0$ such that $\|y\|_{2} \geqslant c\|y\|_{1}$ for all $y \in Y$. Suppose that $Y$ is a Banach space with respect to both norms. Suppose that $T: X \rightarrow Y$ is a linear operator, and that it is bounded when $Y$ is endowed with the $\|\cdot\|_{1}$ norm. Show that it is also bounded when $Y$ is endowed with the $\|\cdot\|_{2}$ norm.

  (ii) Suppose that $X$ is a normed space and that $\left(x_{n}\right)_{n=1}^{\infty} \subseteq X$ is a sequence with $\sum_{n=1}^{\infty}\left|f\left(x_{n}\right)\right|<\infty$ for all $f$ in the dual space $X^{*}$. Show that there is an $M$ such that

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

  for all $f \in X^{*}$.

  (iii) Suppose that $X$ is the space of bounded continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ with the sup norm, and that $Y \subseteq X$ is the subspace of continuously differentiable functions with bounded derivative. Let $T: Y \rightarrow X$ be defined by $T f=f^{\prime}$. Show that the graph of $T$ is closed, but that $T$ is not bounded.

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• Paper 4, Section II, G

  Linear Analysis

  Part II, 2012

  Let $X$ be a Banach space and suppose that $T: X \rightarrow X$ is a bounded linear operator. What is an eigenvalue of $T ?$ What is the spectrum $\sigma(T)$ of $T ?$

  Let $X=C[0,1]$ be the space of continuous real-valued functions $f:[0,1] \rightarrow \mathbb{R}$ endowed with the sup norm. Define an operator $T: X \rightarrow X$ by

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

  where

  $$G(x, y)= \begin{cases}y(x-1) & \text { if } y \leqslant x \\ x(y-1) & \text { if } x \leqslant y\end{cases}$$

  Prove the following facts about $T$ :

  (i) $T f(0)=T f(1)=0$ and the second derivative $(T f)^{\prime \prime}(x)$ is equal to $f(x)$ for $x \in(0,1)$;

  (ii) $T$ is compact;

  (iii) $T$ has infinitely many eigenvalues;

  (iv) 0 is not an eigenvalue of $T$;

  (v) $0 \in \sigma(T)$.

  [The Arzelà-Ascoli theorem may be assumed without proof.]

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• Paper 2, Section II, G

  Linear Analysis

  Part II, 2012

  What is meant by a normal topological space? State and prove Urysohn's lemma.

  Let $X$ be a normal topological space and let $S \subseteq X$ be closed. Show that there is a continuous function $f: X \rightarrow[0,1]$ with $f^{-1}(0)=S$ if, and only if, $S$ is a countable intersection of open sets.

  [Hint. If $S=\bigcap_{n=1}^{\infty} U_{n}$ then consider $\sum_{n=1}^{\infty} 2^{-n} f_{n}$, where the functions $f_{n}: X \rightarrow[0,1]$ are supplied by an appropriate application of Urysohn's lemma.]

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• Paper 1, Section II, G

  Linear Analysis

  Part II, 2012

  What is meant by the dual $X^{*}$ of a normed space $X$ ? Show that $X^{*}$ is a Banach space.

  Let $X=C^{1}(0,1)$, the space of functions $f:(0,1) \rightarrow \mathbb{R}$ possessing a bounded, continuous first derivative. Endow $X$ with the sup norm $\|f\|_{\infty}=\sup _{x \in(0,1)}|f(x)|$. Which of the following maps $T: X \rightarrow \mathbb{R}$ are elements of $X^{*}$ ? Give brief justifications or counterexamples as appropriate.

  1. $T f=f\left(\frac{1}{2}\right)$;
  2. $T f=\|f\|_{\infty}$
  3. $T f=\int_{0}^{1} f(x) d x$
  4. $T f=f^{\prime}\left(\frac{1}{2}\right)$.

  Now suppose that $X$ is a (real) Hilbert space. State and prove the Riesz representation theorem. Describe the natural $\operatorname{map} X \rightarrow X^{* *}$ and show that it is surjective.

  [All normed spaces are over $\mathbb{R}$. You may assume that if $Y$ is a closed subspace of a Hilbert space $X$ then $\left.X=Y \oplus Y^{\perp} .\right]$

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• Paper 3, Section II, F

  Linear Analysis

  Part II, 2013

  State the Stone-Weierstrass Theorem for real-valued functions.

  State Riesz's Lemma.

  Let $K$ be a compact, Hausdorff space and let $A$ be a subalgebra of $C(K)$ separating the points of $K$ and containing the constant functions. Fix two disjoint, non-empty, closed subsets $E$ and $F$ of $K$.

  (i) If $x \in E$ show that there exists $g \in A$ such that $g(x)=0,0 \leqslant g<1$ on $K$, and $g>0$ on $F$. Explain briefly why there is $M \in \mathbb{N}$ such that $g \geqslant \frac{2}{M}$ on $F$.

  (ii) Show that there is an open cover $U_{1}, U_{2}, \ldots, U_{m}$ of $E$, elements $g_{1}, g_{2}, \ldots, g_{m}$ of $A$ and positive integers $M_{1}, M_{2}, \ldots, M_{m}$ such that

  $$0 \leqslant g_{r}<1 \text { on } K, \quad g_{r} \geqslant \frac{2}{M_{r}} \text { on } F, \quad g_{r}<\frac{1}{2 M_{r}} \text { on } U_{r}$$

  for each $r=1,2, \ldots, m$.

  (iii) Using the inequality

  $$1-N t \leqslant(1-t)^{N} \leqslant \frac{1}{N t} \quad(0<t<1, N \in \mathbb{N})$$

  show that for sufficiently large positive integers $n_{1}, n_{2}, \ldots, n_{m}$, the element

  $$h_{r}=1-\left(1-g_{r}^{n_{r}}\right)^{M_{r}^{n_{r}}}$$

  of $A$ satisfies

  $$0 \leqslant h_{r} \leqslant 1 \text { on } K, \quad h_{r} \leqslant \frac{1}{4} \text { on } U_{r}, \quad h_{r} \geqslant\left(\frac{3}{4}\right)^{\frac{1}{m}} \text { on } F$$

  for each $r=1,2, \ldots, m$.

  (iv) Show that the element $h=h_{1} \cdot h_{2} \cdots \cdot h_{m}-\frac{1}{2}$ of $A$ satisfies

  $$-\frac{1}{2} \leqslant h \leqslant \frac{1}{2} \text { on } K, \quad h \leqslant-\frac{1}{4} \text { on } E, \quad h \geqslant \frac{1}{4} \text { on } F \text {. }$$

  Now let $f \in C(K)$ with $\|f\| \leqslant 1$. By considering the sets $\left\{x \in K: f(x) \leqslant-\frac{1}{4}\right\}$ and $\left\{x \in K: f(x) \geqslant \frac{1}{4}\right\}$, show that there exists $h \in A$ such that $\|f-h\| \leqslant \frac{3}{4}$. Deduce that $A$ is dense in $C(K)$.

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• Paper 4, Section II, F

  Linear Analysis

  Part II, 2013

  Let $T: X \rightarrow X$ be a bounded linear operator on a complex Banach space $X$. Define the spectrum $\sigma(T)$ of $T$. What is an approximate eigenvalue of $T$ ? What does it mean to say that $T$ is compact?

  Assume now that $T$ is compact. Show that if $\lambda$ is in the boundary of $\sigma(T)$ and $\lambda \neq 0$, then $\lambda$ is an eigenvalue of $T$. [You may use without proof the result that every $\lambda$ in the boundary of $\sigma(T)$ is an approximate eigenvalue of $T$.]

  Let $T: H \rightarrow H$ be a compact Hermitian operator on a complex Hilbert space $H$. Prove the following:

  (a) If $\lambda \in \sigma(T)$ and $\lambda \neq 0$, then $\lambda$ is an eigenvalue of $T$.

  (b) $\sigma(T)$ is countable.

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• Paper 2, Section II, F

  Linear Analysis

  Part II, 2013

  Let $X$ be a Banach space. Let $T: X \rightarrow \ell_{\infty}$ be a bounded linear operator. Show that there is a bounded sequence $\left(f_{n}\right)_{n=1}^{\infty}$ in $X^{*}$ such that $T x=\left(f_{n} x\right)_{n=1}^{\infty}$ for all $x \in X$.

  Fix $1<p<\infty$. Define the Banach space $\ell_{p}$ and briefly explain why it is separable. Show that for $x \in \ell_{p}$ there exists $f \in \ell_{p}^{*}$ such that $\|f\|=1$ and $f(x)=\|x\|_{p}$. [You may use Hölder's inequality without proof.]

  Deduce that $\ell_{p}$ embeds isometrically into $\ell_{\infty}$.

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• Paper 1, Section II, F

  Linear Analysis

  Part II, 2013

  State and prove the Closed Graph Theorem. [You may assume any version of the Baire Category Theorem provided it is clearly stated. If you use any other result from the course, then you must prove it.]

  Let $X$ be a closed subspace of $\ell_{\infty}$ such that $X$ is also a subset of $\ell_{1}$. Show that the left-shift $L: X \rightarrow \ell_{1}$, given by $L\left(x_{1}, x_{2}, x_{3}, \ldots\right)=\left(x_{2}, x_{3}, \ldots\right)$, is bounded when $X$ is equipped with the sup-norm.

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• Paper 3, Section II, G

  Linear Analysis

  Part II, 2014

  (i) State carefully the theorems of Stone-Weierstrass and Arzelá-Ascoli (work with real scalars only).

  (ii) Let $\mathcal{F}$ denote the family of functions on $[0,1]$ of the form

  $$f(x)=\sum_{n=1}^{\infty} a_{n} \sin (n x)$$

  where the $a_{n}$ are real and $\left|a_{n}\right| \leqslant 1 / n^{3}$ for all $n \in \mathbb{N}$. Prove that any sequence in $\mathcal{F}$ has a subsequence that converges uniformly on $[0,1]$.

  (iii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=0$ and $f^{\prime}(0)$ exists. Show that for each $\varepsilon>0$ there exists a real polynomial $p$ having only odd powers, i.e. of the form

  $$p(x)=a_{1} x+a_{3} x^{3}+\cdots+a_{2 m-1} x^{2 m-1},$$

  such that $\sup _{x \in[0,1]}|f(x)-p(x)|<\varepsilon$. Show that the same holds without the assumption that $f$ is differentiable at 0 .

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• Paper 1, Section II, G

  Linear Analysis

  Part II, 2014

  Let $X$ and $Y$ be normed spaces. What is an isomorphism between $X$ and $Y$ ? Show that a bounded linear map $T: X \rightarrow Y$ is an isomorphism if and only if $T$ is surjective and there is a constant $c>0$ such that $\|T x\| \geqslant c\|x\|$ for all $x \in X$. Show that if there is an isomorphism $T: X \rightarrow Y$ and $X$ is complete, then $Y$ is complete.

  Show that two normed spaces of the same finite dimension are isomorphic. [You may assume without proof that any two norms on a finite-dimensional space are equivalent.] Briefly explain why this implies that every finite-dimensional space is complete, and every closed and bounded subset of a finite-dimensional space is compact.

  Let $Z$ and $F$ be subspaces of a normed space $X$ with $Z \cap F=\{0\}$. Assume that $Z$ is closed in $X$ and $F$ is finite-dimensional. Prove that $Z+F$ is closed in $X$. [Hint: First show that the function $x \mapsto d(x, Z)=\inf \{\|x-z\|: z \in Z\}$ restricted to the unit sphere of F achieves its minimum.]

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• Paper 2, Section II, G

  Linear Analysis

  Part II, 2014

  (a) Let $X$ and $Y$ be Banach spaces, and let $T: X \rightarrow Y$ be a surjective linear map. Assume that there is a constant $c>0$ such that $\|T x\| \geqslant c\|x\|$ for all $x \in X$. Show that $T$ is continuous. [You may use any standard result from general Banach space theory provided you clearly state it.] Give an example to show that the assumption that $X$ and $Y$ are complete is necessary.

  (b) Let $C$ be a closed subset of a Banach space $X$ such that

  (i) $x_{1}+x_{2} \in C$ for each $x_{1}, x_{2} \in C$;

  (ii) $\lambda x \in C$ for each $x \in C$ and $\lambda>0$;

  (iii) for each $x \in X$, there exist $x_{1}, x_{2} \in C$ such that $x=x_{1}-x_{2}$.

  Prove that, for some $M>0$, the unit ball of $X$ is contained in the closure of the set

  $$\left\{x_{1}-x_{2}: x_{i} \in C, \quad\left\|x_{i}\right\| \leqslant M(i=1,2)\right\} .$$

  [You may use without proof any version of the Baire Category Theorem.] Deduce that, for some $K>0$, every $x \in X$ can be written as $x=x_{1}-x_{2}$ with $x_{i} \in C$ and $\left\|x_{i}\right\| \leqslant K\|x\|(i=1,2) .$

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• Paper 4, Section II, G

  Linear Analysis

  Part II, 2014

  Define the spectrum $\sigma(T)$ and the approximate point spectrum $\sigma_{\mathrm{ap}}(T)$ of a bounded linear operator $T$ on a Banach space. Prove that $\sigma_{\mathrm{ap}}(T) \subset \sigma(T)$ and that $\sigma(T)$ is a closed and bounded subset of $\mathbb{C}$. [You may assume without proof that the set of invertible operators is open.]

  Let $T$ be a hermitian operator on a non-zero Hilbert space. Prove that $\sigma(T)$ is not empty

  Let $K$ be a non-empty, compact subset of $\mathbb{C}$. Show that there is a bounded linear operator $T: \ell_{2} \rightarrow \ell_{2}$ with $\sigma(T)=K .$ [You may assume without proof that a compact metric space is separable.]

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• Paper 3, Section II, G

  Linear Analysis

  Part II, 2015

  State and prove the Baire Category Theorem. [Choose any version you like.]

  An isometry from a metric space $(M, d)$ to another metric space $(N, e)$ is a function $\varphi: M \rightarrow N$ such that $e(\varphi(x), \varphi(y))=d(x, y)$ for all $x, y \in M$. Prove that there exists no isometry from the Euclidean plane $\ell_{2}^{2}$ to the Banach space $c_{0}$ of sequences converging to 0 . [Hint: Assume $\varphi: \ell_{2}^{2} \rightarrow \mathrm{c}_{0}$ is an isometry. For $n \in \mathbb{N}$ and $x \in \ell_{2}^{2}$ let $\varphi_{n}(x)$ denote the $n^{\text {th }}$coordinate of $\varphi(x)$. Consider the sets $F_{n}$ consisting of all pairs $(x, y)$ with $\|x\|_{2}=\|y\|_{2}=1$ and $\|\varphi(x)-\varphi(y)\|_{\infty}=\left|\varphi_{n}(x)-\varphi_{n}(y)\right|$.]

  Show that for each $n \in \mathbb{N}$ there is a linear isometry $\ell_{1}^{n} \rightarrow \mathrm{c}_{0}$.

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• Paper 4, Section II, G

  Linear Analysis

  Part II, 2015

  Let $H$ be a Hilbert space and $T \in \mathcal{B}(H)$. Define what is meant by an adjoint of $T$ and prove that it exists, it is linear and bounded, and that it is unique. [You may use the Riesz Representation Theorem without proof.]

  What does it mean to say that $T$ is a normal operator? Give an example of a bounded linear map on $\ell_{2}$ that is not normal.

  Show that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for all $x \in H$.

  Prove that if $T$ is normal, then $\sigma(T)=\sigma_{\mathrm{ap}}(T)$, that is, that every element of the spectrum of $T$ is an approximate eigenvalue of $T$.

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• Paper 2, Section II, G

  Linear Analysis

  Part II, 2015

  (a) Let $T: X \rightarrow Y$ be a linear map between normed spaces. What does it mean to say that $T$ is bounded? Show that $T$ is bounded if and only if $T$ is continuous. Define the operator norm of $T$ and show that the set $\mathcal{B}(X, Y)$ of all bounded, linear maps from $X$ to $Y$ is a normed space in the operator norm.

  (b) For each of the following linear maps $T$, determine if $T$ is bounded. When $T$ is bounded, compute its operator norm and establish whether $T$ is compact. Justify your answers. Here $\|f\|_{\infty}=\sup _{t \in[0,1]}|f(t)|$ for $f \in C[0,1]$ and $\|f\|=\|f\|_{\infty}+\left\|f^{\prime}\right\|_{\infty}$ for $f \in C^{1}[0,1]$.

  (i) $T:\left(C^{1}[0,1],\|\cdot\|_{\infty}\right) \rightarrow\left(C^{1}[0,1],\|\cdot\|\right), T(f)=f$.

  (ii) $T:\left(C^{1}[0,1],\|\cdot\|\right) \rightarrow\left(C[0,1],\|\cdot\|_{\infty}\right), T(f)=f$.

  (iii) $T:\left(C^{1}[0,1],\|\cdot\|\right) \rightarrow\left(C[0,1],\|\cdot\|_{\infty}\right), T(f)=f^{\prime}$.

  (iv) $T:\left(C[0,1],\|\cdot\|_{\infty}\right) \rightarrow \mathbb{R}, T(f)=\int_{0}^{1} f(t) h(t) d t$, where $h$ is a given element of $C[0,1]$. [Hint: Consider first the case that $h(x) \neq 0$ for every $x \in[0,1]$, and apply $T$ to a suitable function. In the general case apply $T$ to a suitable sequence of functions.]

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• Paper 1, Section II, G

  Linear Analysis

  Part II, 2015

  (a) Let $\left(e_{n}\right)_{n=1}^{\infty}$ be an orthonormal basis of an inner product space $X$. Show that for all $x \in X$ there is a unique sequence $\left(a_{n}\right)_{n=1}^{\infty}$ of scalars such that $x=\sum_{n=1}^{\infty} a_{n} e_{n}$.

  Assume now that $X$ is a Hilbert space and that $\left(f_{n}\right)_{n=1}^{\infty}$ is another orthonormal basis of $X$. Prove that there is a unique bounded linear map $U: X \rightarrow X$ such that $U\left(e_{n}\right)=f_{n}$ for all $n \in \mathbb{N}$. Prove that this map $U$ is unitary.

  (b) Let $1 \leqslant p<\infty$ with $p \neq 2$. Show that no subspace of $\ell_{2}$ is isomorphic to $\ell_{p}$. [Hint: Apply the generalized parallelogram law to suitable vectors.]

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• Paper 3, Section II, I

  Linear Analysis

  Part II, 2016

  (a) Define Banach spaces and Euclidean spaces over $\mathbb{R}$. [You may assume the definitions of vector spaces and inner products.]

  (b) Let $X$ be the space of sequences of real numbers with finitely many non-zero entries. Does there exist a norm $\|\cdot\|$ on $X$ such that $(X,\|\cdot\|)$ is a Banach space? Does there exist a norm such that $(X,\|\cdot\|)$ is Euclidean? Justify your answers.

  (c) Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{R}$ satisfying the parallelogram law

  $$\|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}$$

  for all $x, y \in X$. Show that $\langle x, y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)$ is an inner product on $X$. [You may use without proof the fact that the vector space operations $+$ and are continuous with respect to $\|\cdot\|$. To verify the identity $\langle a+b, c\rangle=\langle a, c\rangle+\langle b, c\rangle$, you may find it helpful to consider the parallelogram law for the pairs $(a+c, b),(b+c, a),(a-c, b)$ and $(b-c, a) .]$

  (d) Let $\left(X,\|\cdot\|_{X}\right)$ be an incomplete normed vector space over $\mathbb{R}$ which is not a Euclidean space, and let $\left(X^{*},\|\cdot\|_{X^{*}}\right)$ be its dual space with the dual norm. Is $\left(X^{*},\|\cdot\|_{X^{*}}\right)$ a Banach space? Is it a Euclidean space? Justify your answers.

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• Paper 2, Section II, I

  Linear Analysis

  Part II, 2016

  (a) Let $K$ be a topological space and let $C_{\mathbb{R}}(K)$ denote the normed vector space of bounded continuous real-valued functions on $K$ with the norm $\|f\|_{C_{\mathbb{R}}(K)}=\sup _{x \in K}|f(x)|$. Define the terms uniformly bounded, equicontinuous and relatively compact as applied to subsets $S \subset C_{\mathbb{R}}(K)$.

  (b) The Arzela-Ascoli theorem [which you need not prove] states in particular that if $K$ is compact and $S \subset C_{\mathbb{R}}(K)$ is uniformly bounded and equicontinuous, then $S$ is relatively compact. Show by examples that each of the compactness of $K$, uniform boundedness of $S$, and equicontinuity of $S$ are necessary conditions for this conclusion.

  (c) Let $L$ be a topological space. Assume that there exists a sequence of compact subsets $K_{n}$ of $L$ such that $K_{1} \subset K_{2} \subset K_{3} \subset \cdots \subset L$ and $\bigcup_{n=1}^{\infty} K_{n}=L$. Suppose $S \subset C_{\mathbb{R}}(L)$ is uniformly bounded and equicontinuous and moreover satisfies the condition that, for every $\epsilon>0$, there exists $n \in \mathbb{N}$ such that $|f(x)|<\epsilon$ for every $x \in L \backslash K_{n}$ and for every $f \in S$. Show that $S$ is relatively compact.

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• Paper 1, Section II, I

  Linear Analysis

  Part II, 2016

  (a) State the closed graph theorem.

  (b) Prove the closed graph theorem assuming the inverse mapping theorem.

  (c) Let $X, Y, Z$ be Banach spaces and $T: X \rightarrow Y, S: Y \rightarrow Z$ be linear maps. Suppose that $S \circ T$ is bounded and $S$ is both bounded and injective. Show that $T$ is bounded.

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• Paper 4, Section II, I

  Linear Analysis

  Part II, 2016

  Let $H$ be a complex Hilbert space.

  (a) Let $T: H \rightarrow H$ be a bounded linear map. Show that the spectrum of $T$ is a subset of $\left\{\lambda \in \mathbb{C}:|\lambda| \leqslant\|T\|_{\mathcal{B}(H)}\right\}$.

  (b) Let $T: H \rightarrow H$ be a bounded self-adjoint linear map. For $\lambda, \mu \in \mathbb{C}$, let $E_{\lambda}:=\{x \in H: T x=\lambda x\}$ and $E_{\mu}:=\{x \in H: T x=\mu x\}$. If $\lambda \neq \mu$, show that $E_{\lambda} \perp E_{\mu}$.

  (c) Let $T: H \rightarrow H$ be a compact self-adjoint linear map. For $\lambda \neq 0$, show that $E_{\lambda}:=\{x \in H: T x=\lambda x\}$ is finite-dimensional.

  (d) Let $H_{1} \subset H$ be a closed, proper, non-trivial subspace. Let $P$ be the orthogonal projection to $H_{1}$.

  (i) Prove that $P$ is self-adjoint.

  (ii) Determine the spectrum $\sigma(P)$ and the point spectrum $\sigma_{p}(P)$ of $P$.

  (iii) Find a necessary and sufficient condition on $H_{1}$ for $P$ to be compact.

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• Paper 3, Section II, F

  Linear Analysis

  Part II, 2017

  Let $K$ be a non-empty compact Hausdorff space and let $C(K)$ be the space of real-valued continuous functions on $K$.

  (i) State the real version of the Stone-Weierstrass theorem.

  (ii) Let $A$ be a closed subalgebra of $C(K)$. Prove that $f \in A$ and $g \in A$ implies that $m \in A$ where the function $m: K \rightarrow \mathbb{R}$ is defined by $m(x)=\max \{f(x), g(x)\}$. [You may use without proof that $f \in A$ implies $|f| \in A$.]

  (iii) Prove that $K$ is normal and state Urysohn's Lemma.

  (iv) For any $x \in K$, define $\delta_{x} \in C(K)^{*}$ by $\delta_{x}(f)=f(x)$ for $f \in C(K)$. Justifying your answer carefully, find

  $$\inf _{x \neq y}\left\|\delta_{x}-\delta_{y}\right\| .$$

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• Paper 2, Section II, F

  Linear Analysis

  Part II, 2017

  (a) Let $X$ be a normed vector space and $Y \subset X$ a closed subspace with $Y \neq X$. Show that $Y$ is nowhere dense in $X$.

  (b) State any version of the Baire Category theorem.

  (c) Let $X$ be an infinite-dimensional Banach space. Show that $X$ cannot have a countable algebraic basis, i.e. there is no countable subset $\left(x_{k}\right)_{k \in \mathbb{N}} \subset X$ such that every $x \in X$ can be written as a finite linear combination of elements of $\left(x_{k}\right)$.

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• Paper 1, Section II, F

  Linear Analysis

  Part II, 2017

  Let $X$ be a normed vector space over the real numbers.

  (a) Define the dual space $X^{*}$ of $X$ and prove that $X^{*}$ is a Banach space. [You may use without proof that $X^{*}$ is a vector space.]

  (b) The Hahn-Banach theorem states the following. Let $X$ be a real vector space, and let $p: X \rightarrow \mathbb{R}$ be sublinear, i.e., $p(x+y) \leqslant p(x)+p(y)$ and $p(\lambda x)=\lambda p(x)$ for all $x, y \in X$ and all $\lambda>0$. Let $Y \subset X$ be a linear subspace, and let $g: Y \rightarrow \mathbb{R}$ be linear and satisfy $g(y) \leqslant p(y)$ for all $y \in Y$. Then there exists a linear functional $f: X \rightarrow \mathbb{R}$ such that $f(x) \leqslant p(x)$ for all $x \in X$ and $\left.f\right|_{Y}=g$.

  Using the Hahn-Banach theorem, prove that for any non-zero $x_{0} \in X$ there exists $f \in X^{*}$ such that $f\left(x_{0}\right)=\left\|x_{0}\right\|$ and $\|f\|=1$.

  (c) Show that $X$ can be embedded isometrically into a Banach space, i.e. find a Banach space $Y$ and a linear map $\Phi: X \rightarrow Y$ with $\|\Phi(x)\|=\|x\|$ for all $x \in X$.

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• Paper 4, Section II, F

  Linear Analysis

  Part II, 2017

  Let $H$ be a complex Hilbert space with inner product $(\cdot, \cdot)$ and let $T: H \rightarrow H$ be a bounded linear map.

  (i) Define the spectrum $\sigma(T)$, the point spectrum $\sigma_{p}(T)$, the continuous spectrum $\sigma_{c}(T)$, and the residual spectrum $\sigma_{r}(T)$.

  (ii) Show that $T^{*} T$ is self-adjoint and that $\sigma\left(T^{*} T\right) \subset[0, \infty)$. Show that if $T$ is compact then so is $T^{*} T$.

  (iii) Assume that $T$ is compact. Prove that $T$ has a singular value decomposition: for $N<\infty$ or $N=\infty$, there exist orthonormal systems $\left(u_{i}\right)_{i=1}^{N} \subset H$ and $\left(v_{i}\right)_{i=1}^{N} \subset H$ and $\left(\lambda_{i}\right)_{i=1}^{N} \subset[0, \infty)$ such that, for any $x \in H$,

  $$T x=\sum_{i=1}^{N} \lambda_{i}\left(u_{i}, x\right) v_{i}$$

  [You may use the spectral theorem for compact self-adjoint linear operators.]

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• Paper 3, Section II, F

  Linear Analysis

  Part II, 2018

  (a) Let $X$ be a normed vector space and let $Y$ be a Banach space. Show that the space of bounded linear operators $\mathcal{B}(X, Y)$ is a Banach space.

  (b) Let $X$ and $Y$ be Banach spaces, and let $D \subset X$ be a dense linear subspace. Prove that a bounded linear map $T: D \rightarrow Y$ can be extended uniquely to a bounded linear map $T: X \rightarrow Y$ with the same operator norm. Is the claim also true if one of $X$ and $Y$ is not complete?

  (c) Let $X$ be a normed vector space. Let $\left(x_{n}\right)$ be a sequence in $X$ such that

  $$\sum_{n=1}^{\infty}\left|f\left(x_{n}\right)\right|<\infty \quad \forall f \in X^{*}$$

  Prove that there is a constant $C$ such that

  $$\sum_{n=1}^{\infty}\left|f\left(x_{n}\right)\right| \leqslant C\|f\| \quad \forall f \in X^{*}$$

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• Paper 1, Section II, F

  Linear Analysis

  Part II, 2018

  Let $K$ be a compact Hausdorff space.

  (a) State the Arzelà-Ascoli theorem, and state both the real and complex versions of the Stone-Weierstraß theorem. Give an example of a compact space $K$ and a bounded set of functions in $C(K)$ that is not relatively compact.

  (b) Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be continuous. Show that there exists a sequence of polynomials $\left(p_{i}\right)$ in $n$ variables such that

  $$B \subset \mathbb{R}^{n} \text { compact }\left.\left.\Rightarrow \quad p_{i}\right|_{B} \rightarrow f\right|_{B} \text { uniformly }$$

  Characterize the set of continuous functions $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ for which there exists a sequence of polynomials $\left(p_{i}\right)$ such that $p_{i} \rightarrow f$ uniformly on $\mathbb{R}^{n}$.

  (c) Prove that if $C(K)$ is equicontinuous then $K$ is finite. Does this implication remain true if we drop the requirement that $K$ be compact? Justify your answer.

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• Paper 2, Section II, F

  Linear Analysis

  Part II, 2018

  Let $X, Y$ be Banach spaces and let $\mathcal{B}(X, Y)$ denote the space of bounded linear operators $T: X \rightarrow Y$.

  (a) Define what it means for a bounded linear operator $T: X \rightarrow Y$ to be compact. Let $T_{i}: X \rightarrow Y$ be linear operators with finite rank, i.e., $T_{i}(X)$ is finite-dimensional. Assume that the sequence $T_{i}$ converges to $T$ in $\mathcal{B}(X, Y)$. Show that $T$ is compact.

  (b) Let $T: X \rightarrow Y$ be compact. Show that the dual map $T^{*}: Y^{*} \rightarrow X^{*}$ is compact. [Hint: You may use the Arzelà-Ascoli theorem.]

  (c) Let $X$ be a Hilbert space and let $T: X \rightarrow X$ be a compact operator. Let $\left(\lambda_{j}\right)$ be an infinite sequence of eigenvalues of $T$ with eigenvectors $x_{j}$. Assume that the eigenvectors are orthogonal to each other. Show that $\lambda_{j} \rightarrow 0$.

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• Paper 4, Section II, F

  Linear Analysis

  Part II, 2018

  (a) Let $X$ be a separable normed space. For any sequence $\left(f_{n}\right)_{n \in \mathbb{N}} \subset X^{*}$ with $\left\|f_{n}\right\| \leqslant 1$ for all $n$, show that there is $f \in X^{*}$ and a subsequence $\Lambda \subset \mathbb{N}$ such that $f_{n}(x) \rightarrow f(x)$ for all $x \in X$ as $n \in \Lambda, n \rightarrow \infty$. [You may use without proof the fact that $X^{*}$ is complete and that any bounded linear map $f: D \rightarrow \mathbb{R}$, where $D \subset X$ is a dense linear subspace, can be extended uniquely to an element $f \in X^{*}$.]

  (b) Let $H$ be a Hilbert space and $U: H \rightarrow H$ a unitary map. Let

  $$I=\{x \in H: U x=x\}, \quad W=\{U x-x: x \in H\}$$

  Prove that $I$ and $W$ are orthogonal, $H=I \oplus \bar{W}$, and that for every $x \in H$,

  $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=0}^{n-1} U^{i} x=P x$$

  where $P$ is the orthogonal projection onto the closed subspace $I$.

  (c) Let $T: C\left(S^{1}\right) \rightarrow C\left(S^{1}\right)$ be a linear map, where $S^{1}=\left\{e^{i \theta} \in \mathbb{C}: \theta \in \mathbb{R}\right\}$ is the unit circle, induced by a homeomorphism $\tau: S^{1} \rightarrow S^{1}$ by $(T f) e^{i \theta}=f\left(\tau\left(e^{i \theta}\right)\right)$. Prove that there exists $\mu \in C\left(S^{1}\right)^{*}$ with $\mu\left(1_{S^{1}}\right)=1$ such that $\mu(T f)=\mu(f)$ for all $f \in C\left(S^{1}\right)$. (Here $1_{S^{1}}$ denotes the function on $S^{1}$ which returns 1 identically.) If $T$ is not the identity map, does it follow that $\mu$ as above is necessarily unique? Justify your answer.

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• Paper 3, Section II, H

  Linear Analysis

  Part II, 2019

  (a) Let $X$ be a Banach space and consider the open unit ball $B=\{x \in X:\|x\|<1\}$. Let $T: X \rightarrow X$ be a bounded operator. Prove that $\overline{T(B)} \supset B \operatorname{implies} T(B) \supset B$.

  (b) Let $P$ be the vector space of all polynomials in one variable with real coefficients. Let $\|\cdot\|$ be any norm on $P$. Show that $(P,\|\cdot\|)$ is not complete.

  (c) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be entire, and assume that for every $z \in \mathbb{C}$ there is $n$ such that $f^{(n)}(z)=0$ where $f^{(n)}$ is the $n$-th derivative of $f$. Prove that $f$ is a polynomial.

  [You may use that an entire function vanishing on an open subset of $\mathbb{C}$ must vanish everywhere.]

  (d) A Banach space $X$ is said to be uniformly convex if for every $\varepsilon \in(0,2]$ there is $\delta>0$ such that for all $x, y \in X$ such that $\|x\|=\|y\|=1$ and $\|x-y\| \geqslant \varepsilon$, one has $\|(x+y) / 2\| \leqslant 1-\delta$. Prove that $\ell^{2}$ is uniformly convex.

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• Paper 4, Section II, H

  Linear Analysis

  Part II, 2019

  (a) State and prove the Riesz representation theorem for a real Hilbert space $H$.

  [You may use that if $H$ is a real Hilbert space and $Y \subset H$ is a closed subspace, then $\left.H=Y \oplus Y_{.}^{\perp} .\right]$

  (b) Let $H$ be a real Hilbert space and $T: H \rightarrow H$ a bounded linear operator. Show that $T$ is invertible if and only if both $T$ and $T^{*}$ are bounded below. [Recall that an operator $S: H \rightarrow H$ is bounded below if there is $c>0$ such that $\|S x\| \geqslant c\|x\|$ for all $x \in H$.]

  (c) Consider the complex Hilbert space of two-sided sequences,

  $$X=\left\{\left(x_{n}\right)_{n \in \mathbb{Z}}: x_{n} \in \mathbb{C}, \sum_{n \in \mathbb{Z}}\left|x_{n}\right|^{2}<\infty\right\}$$

  with norm $\|x\|=\left(\sum_{n}\left|x_{n}\right|^{2}\right)^{1 / 2}$. Define $T: X \rightarrow X$ by $(T x)_{n}=x_{n+1}$. Show that $T$ is unitary and find the point spectrum and the approximate point spectrum of $T$.

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• Paper 2, Section II, H

  Linear Analysis

  Part II, 2019

  (a) State the real version of the Stone-Weierstrass theorem and state the UrysohnTietze extension theorem.

  (b) In this part, you may assume that there is a sequence of polynomials $P_{i}$ such that $\sup _{x \in[0,1]}\left|P_{i}(x)-\sqrt{x}\right| \rightarrow 0$ as $i \rightarrow \infty$.

  Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous piecewise linear function which is linear on $[0,1 / 2]$ and on $[1 / 2,1]$. Using the polynomials $P_{i}$ mentioned above (but not assuming any form of the Stone-Weierstrass theorem), prove that there are polynomials $Q_{i}$ such that $\sup _{x \in[0,1]}\left|Q_{i}(x)-f(x)\right| \rightarrow 0$ as $i \rightarrow \infty$.

  (d) Which of the following families of functions are relatively compact in $C[0,1]$ with the supremum norm? Justify your answer.

  $$\begin{aligned} &\mathcal{F}_{1}=\left\{x \mapsto \frac{\sin (\pi n x)}{n}: n \in \mathbb{N}\right\} \\ &\mathcal{F}_{2}=\left\{x \mapsto \frac{\sin (\pi n x)}{n^{1 / 2}}: n \in \mathbb{N}\right\} \\ &\mathcal{F}_{3}=\{x \mapsto \sin (\pi n x): n \in \mathbb{N}\} \end{aligned}$$

  [In this question $\mathbb{N}$ denotes the set of positive integers.]

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• Paper 1, Section II, H

  Linear Analysis

  Part II, 2019

  Let $F$ be the space of real-valued sequences with only finitely many nonzero terms.

  (a) For any $p \in[1, \infty)$, show that $F$ is dense in $\ell^{p}$. Is $F$ dense in $\ell^{\infty} ?$ Justify your answer.

  (b) Let $p \in[1, \infty)$, and let $T: F \rightarrow F$ be an operator that is bounded in the $\|\cdot\|_{p}$-norm, i.e., there exists a $C$ such that $\|T x\|_{p} \leqslant C\|x\|_{p}$ for all $x \in F$. Show that there is a unique bounded operator $\widetilde{T}: \ell^{p} \rightarrow \ell^{p}$ satisfying $\left.\widetilde{T}\right|_{F}=T$, and that $\|\widetilde{T}\|_{p} \leqslant C$.

  (c) For each $p \in[1, \infty]$ and for each $i=1, \ldots, 5$ determine if there is a bounded operator from $\ell^{p}$ to $\ell^{p}$ (in the $\|\cdot\|_{p}$ norm) whose restriction to $F$ is given by $T_{i}$ :

  $$\begin{gathered} \left(T_{1} x\right)_{n}=n x_{n}, \quad\left(T_{2} x\right)_{n}=n\left(x_{n}-x_{n+1}\right), \quad\left(T_{3} x\right)_{n}=\frac{x_{n}}{n}, \\ \left(T_{4} x\right)_{n}=\frac{x_{1}}{n^{1 / 2}}, \quad\left(T_{5} x\right)_{n}=\frac{\sum_{j=1}^{n} x_{j}}{2^{n}} \end{gathered}$$

  (d) Let $X$ be a normed vector space such that the closed unit ball $\overline{B_{1}(0)}$ is compact. Prove that $X$ is finite dimensional.

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• Paper 1, Section II, I

  Linear Analysis

  Part II, 2020

  (a) Define the dual space $X^{*}$ of a (real) normed space $(X,\|\cdot\|)$. Define what it means for two normed spaces to be isometrically isomorphic. Prove that $\left(l_{1}\right)^{*}$ is isometrically isomorphic to $l_{\infty}$.

  (b) Let $p \in(1, \infty)$. [In this question, you may use without proof the fact that $\left(l_{p}\right)^{*}$ is isometrically isomorphic to $l_{q}$ where $\frac{1}{p}+\frac{1}{q}=1$.]

  (i) Show that if $\left\{\phi_{m}\right\}_{m=1}^{\infty}$ is a countable dense subset of $\left(l_{p}\right)^{*}$, then the function

  $$d(x, y):=\sum_{m=1}^{\infty} 2^{-m} \frac{\left|\phi_{m}(x-y)\right|}{1+\left|\phi_{m}(x-y)\right|}$$

  defines a metric on the closed unit ball $B \subset l_{p}$. Show further that for a sequence $\left\{x^{(n)}\right\}_{n=1}^{\infty}$ of elements $x^{(n)} \in B$, we have

  $$\phi\left(x^{(n)}\right) \rightarrow \phi(x) \quad \forall \phi \in\left(l_{p}\right)^{*} \quad \Leftrightarrow \quad d\left(x^{(n)}, x\right) \rightarrow 0$$

  Deduce that $(B, d)$ is a compact metric space.

  (ii) Give an example to show that for a sequence $\left\{x^{(n)}\right\}_{n=1}^{\infty}$ of elements $x^{(n)} \in B$ and $x \in B$,

  $$\phi\left(x^{(n)}\right) \rightarrow \phi(x) \quad \forall \phi \in\left(l_{p}\right)^{*} \quad \Rightarrow \quad\left\|x^{(n)}-x\right\|_{l_{p}} \rightarrow 0$$

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• Paper 2, Section II, I

  Linear Analysis

  Part II, 2020

  (a) State and prove the Baire Category theorem.

  Let $p>1$. Apply the Baire Category theorem to show that $\bigcup_{1 \leqslant q<p} l_{q} \neq l_{p}$. Give an explicit element of $l_{p} \backslash \bigcup_{1 \leqslant q<p} l_{q}$.

  (b) Use the Baire Category theorem to prove that $C([0,1])$ contains a function which is nowhere differentiable.

  (c) Let $(X,\|\cdot\|)$ be a real Banach space. Verify that the map sending $x$ to the function $e_{x}: \phi \mapsto \phi(x)$ is a continuous linear map of $X$ into $\left(X^{*}\right)^{*}$ where $X^{*}$ denotes the dual space of $(X,\|\cdot\|)$. Taking for granted the fact that this map is an isometry regardless of the norm on $X$, prove that if $\|\cdot\|^{\prime}$ is another norm on the vector space $X$ which is not equivalent to $\|\cdot\|$, then there is a linear function $\psi: X \rightarrow \mathbb{R}$ which is continuous with respect to one of the two norms $\|\cdot\|,\|\cdot\|^{\prime}$ and not continuous with respect to the other.

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• Paper 3, Section II, I

  Linear Analysis

  Part II, 2020

  Let $H$ be a separable complex Hilbert space.

  (a) For an operator $T: H \rightarrow H$, define the spectrum and point spectrum. Define what it means for $T$ to be: (i) a compact operator; (ii) a self-adjoint operator and (iii) a finite rank operator.

  (b) Suppose $T: H \rightarrow H$ is compact. Prove that given any $\delta>0$, there exists a finite-dimensional subspace $E \subset H$ such that $\left\|T\left(e_{n}\right)-P_{E} T\left(e_{n}\right)\right\|<\delta$ for each $n$, where $\left\{e_{1}, e_{2}, e_{3}, \ldots\right\}$ is an orthonormal basis for $H$ and $P_{E}$ denotes the orthogonal projection onto $E$. Deduce that a compact operator is the operator norm limit of finite rank operators.

  (c) Suppose that $S: H \rightarrow H$ has finite rank and $\lambda \in \mathbb{C} \backslash\{0\}$ is not an eigenvalue of $S$. Prove that $S-\lambda I$ is surjective. [You may wish to consider the action of $S(S-\lambda I)$ on $\left.\operatorname{ker}(S)^{\perp} .\right]$

  (d) Suppose $T: H \rightarrow H$ is compact and $\lambda \in \mathbb{C} \backslash\{0\}$ is not an eigenvalue of $T$. Prove that the image of $T-\lambda I$ is dense in $H$.

  Prove also that $T-\lambda I$ is bounded below, i.e. prove also that there exists a constant $c>0$ such that $\|(T-\lambda I) x\| \geqslant c\|x\|$ for all $x \in H$. Deduce that $T-\lambda I$ is surjective.

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• Paper 4, Section II, I

  Linear Analysis

  Part II, 2020

  (a) For $K$ a compact Hausdorff space, what does it mean to say that a set $S \subset C(K)$ is equicontinuous. State and prove the Arzelà-Ascoli theorem.

  (b) Suppose $K$ is a compact Hausdorff space for which $C(K)$ is a countable union of equicontinuous sets. Prove that $K$ is finite.

  (c) Let $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a bounded, continuous function and let $x_{0} \in \mathbb{R}^{n}$. Consider the problem of finding a differentiable function $x:[0,1] \rightarrow \mathbb{R}^{n}$ with

  $$x(0)=x_{0} \quad \text { and } \quad x^{\prime}(t)=F(x(t))$$

  for all $t \in[0,1]$. For each $k=1,2,3, \ldots$, let $x_{k}:[0,1] \rightarrow \mathbb{R}^{n}$ be defined by setting $x_{k}(0)=x_{0}$ and

  $$x_{k}(t)=x_{0}+\int_{0}^{t} F\left(y_{k}(s)\right) d s$$

  for $t \in[0,1]$, where

  $$y_{k}(t)=x_{k}\left(\frac{j}{k}\right)$$

  for $t \in\left(\frac{j}{k}, \frac{j+1}{k}\right]$ and $j \in\{0,1, \ldots, k-1\}$.

  (i) Verify that $x_{k}$ is well-defined and continuous on $[0,1]$ for each $k$.

  (ii) Prove that there exists a differentiable function $x:[0,1] \rightarrow \mathbb{R}^{n}$ solving (*) for $t \in[0,1]$.

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• Paper 1, Section II, 22H

  Linear Analysis

  Part II, 2021

  Let $H$ be a separable Hilbert space and $\left\{e_{i}\right\}$ be a Hilbertian (orthonormal) basis of $H$. Given a sequence $\left(x_{n}\right)$ of elements of $H$ and $x_{\infty} \in H$, we say that $x_{n}$ weakly converges to $x_{\infty}$, denoted $x_{n} \rightarrow x_{\infty}$, if $\forall h \in H, \lim _{n \rightarrow \infty}\left\langle x_{n}, h\right\rangle=\left\langle x_{\infty}, h\right\rangle$.

  (a) Given a sequence $\left(x_{n}\right)$ of elements of $H$, prove that the following two statements are equivalent:

  (i) $\exists x_{\infty} \in H$ such that $x_{n} \rightarrow x_{\infty}$;

  (ii) the sequence $\left(x_{n}\right)$ is bounded in $H$ and $\forall i \geqslant 1$, the sequence $\left(\left\langle x_{n}, e_{i}\right\rangle\right)$ is convergent.

  (b) Let $\left(x_{n}\right)$ be a bounded sequence of elements of $H$. Show that there exists $x_{\infty} \in H$ and a subsequence $\left(x_{\phi(n)}\right)$ such that $x_{\phi(n)} \rightarrow x_{\infty}$ in $H$.

  (c) Let $\left(x_{n}\right)$ be a sequence of elements of $H$ and $x_{\infty} \in H$ be such that $x_{n} \rightarrow x_{\infty}$. Show that the following three statements are equivalent:

  (i) $\lim _{n \rightarrow \infty}\left\|x_{n}-x_{\infty}\right\|=0$;

  (ii) $\lim _{n \rightarrow \infty}\left\|x_{n}\right\|=\left\|x_{\infty}\right\|$;

  (iii) $\forall \epsilon>0, \exists I(\epsilon)$ such that $\forall n \geqslant 1, \sum_{i \geqslant I(\epsilon)}\left|\left\langle x_{n}, e_{i}\right\rangle\right|^{2}<\epsilon$.

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• Paper 2, Section II, 22H

  Linear Analysis

  Part II, 2021

  (a) Let $V$ be a real normed vector space. Show that any proper subspace of $V$ has empty interior.

  Assuming $V$ to be infinite-dimensional and complete, prove that any algebraic basis of $V$ is uncountable. [The Baire category theorem can be used if stated properly.] Deduce that the vector space of polynomials with real coefficients cannot be equipped with a complete norm, i.e. a norm that makes it complete.

  (b) Suppose that $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ are norms on a vector space $V$ such that $\left(V,\|\cdot\|_{1}\right)$ and $\left(V,\|\cdot\|_{2}\right)$ are both complete. Prove that if there exists $C_{1}>0$ such that $\|x\|_{2} \leqslant C_{1}\|x\|_{1}$ for all $x \in V$, then there exists $C_{2}>0$ such that $\|x\|_{1} \leqslant C_{2}\|x\|_{2}$ for all $x \in V$. Is this still true without the assumption that $\left(V,\|\cdot\|_{1}\right)$ and $\left(V,\|\cdot\|_{2}\right)$ are both complete? Justify your answer.

  (c) Let $V$ be a real normed vector space (not necessarily complete) and $V^{*}$ be the set of linear continuous forms $f: V \rightarrow \mathbb{R}$. Let $\left(x_{n}\right)_{n \geqslant 1}$ be a sequence in $V$ such that $\sum_{n \geqslant 1}\left|f\left(x_{n}\right)\right|<\infty$ for all $f \in V^{*}$. Prove that

  $$\sup _{\|f\|_{V^{*} \leqslant 1}} \sum_{n \geqslant 1}\left|f\left(x_{n}\right)\right|<\infty .$$

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• Paper 3, Section II, H

  Linear Analysis

  Part II, 2021

  (a) State the Arzela-Ascoli theorem, including the definition of equicontinuity.

  (b) Consider a sequence $\left(f_{n}\right)$ of continuous real-valued functions on $\mathbb{R}$ such that for all $x \in \mathbb{R},\left(f_{n}(x)\right)$ is bounded and the sequence is equicontinuous at $x$. Prove that there exists $f \in C(\mathbb{R})$ and a subsequence $\left(f_{\varphi(n)}\right)$ such that $f_{\varphi(n)} \rightarrow f$ uniformly on any closed bounded interval.

  (c) Let $K$ be a Hausdorff compact topological space, and $C(K)$ the real-valued continuous functions on $K$. Let $\mathcal{K} \subset C(K)$ be a compact subset of $C(K)$. Prove that the collection of functions $\mathcal{K}$ is equicontinuous.

  (d) We say that a Hausdorff topological space $X$ is locally compact if every point has a compact neighbourhood. Let $X$ be such a space, $K \subset X$ compact and $U \subset X$ open such that $K \subset U$. Prove that there exists $f: X \rightarrow \mathbb{R}$ continuous with compact support contained in $U$ and equal to 1 on $K$. [Hint: Construct an open set $V$ such that $K \subset V \subset \bar{V} \subset U$ and $\bar{V}$ is compact, and use Urysohn's lemma to construct a function in $\bar{V}$ and then extend it by zero.]

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• Paper 4, Section II, H

  Linear Analysis

  Part II, 2021

  (a) Let $\left(H_{1},\langle\cdot, \cdot\rangle_{1}\right),\left(H_{2},\langle\cdot, \cdot\rangle_{2}\right)$ be two Hilbert spaces, and $T: H_{1} \rightarrow H_{2}$ be a bounded linear operator. Show that there exists a unique bounded linear operator $T^{*}: H_{2} \rightarrow H_{1}$ such that

  $$\left\langle T x_{1}, x_{2}\right\rangle_{2}=\left\langle x_{1}, T^{*} x_{2}\right\rangle_{1}, \quad \forall x_{1} \in H_{1}, x_{2} \in H_{2}$$

  (b) Let $H$ be a separable Hilbert space. We say that a sequence $\left(e_{i}\right)$ is a frame of $H$ if there exists $A, B>0$ such that

  $$\forall x \in H, \quad A\|x\|^{2} \leqslant \sum_{i \geqslant 1}\left|\left\langle x, e_{i}\right\rangle\right|^{2} \leqslant B\|x\|^{2}$$

  State briefly why such a frame exists. From now on, let $\left(e_{i}\right)$ be a frame of $H$. Show that $\operatorname{Span}\left\{e_{i}\right\}$ is dense in $H$.

  (c) Show that the linear map $U: H \rightarrow \ell^{2}$ given by $U(x)=\left(\left\langle x, e_{i}\right\rangle\right)_{i \geqslant 1}$ is bounded and compute its adjoint $U^{*}$.

  (d) Assume now that $\left(e_{i}\right)$ is a Hilbertian (orthonormal) basis of $H$ and let $a \in H$. Show that the Hilbert cube $\mathcal{C}_{a}=\left\{x \in H\right.$ such that $\left.\forall i \geqslant 1,\left|\left\langle x, e_{i}\right\rangle\right| \leqslant\left|\left\langle a, e_{i}\right\rangle\right|\right\}$ is a compact subset of $H$.

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