# Linear Analysis

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Paper 1, Section II, 22H

commentLet $H$ 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 $\forall n \geqslant 1, \sum_{i \geqslant I(\epsilon)}\left|\left\langle x_{n}, e_{i}\right\rangle\right|^{2}<\epsilon$.

Paper 2, Section II, 22H

comment(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

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

Paper 3, Section II, H

comment(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.]

Paper 4, Section II, H

comment(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 $H$.

Paper 1, Section II, I

comment(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 $x \in B$,

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

Paper 2, Section II, I

comment(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.

Paper 3, Section II, I

commentLet $H$ 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.

Paper 4, Section II, I

comment(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 $t \in[0,1]$.

Paper 1, Section II, H

commentLet $F$ 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.

Paper 2, Section II, H

comment(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.]

Paper 3, Section II, H

comment(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.

Paper 4, Section II, H

comment(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 $T$.

Paper 1, Section II, F

commentLet $K$ 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.

Paper 2, Section II, F

commentLet $X, Y$ 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 $\lambda_{j} \rightarrow 0$.

Paper 3, Section II, F

comment(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

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

Paper 4, Section II, F

comment(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.

Paper 1, Section II, F

commentLet $X$ 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 $x \in X$.

Paper 2, Section II, F

comment(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 $\left(x_{k}\right)$.

Paper 3, Section II, F

commentLet $K$ 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

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

Paper 4, Section II, F

commentLet $H$ 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.]

Paper 1, Section II, I

comment(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.

Paper 2, Section II, I

comment(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.

Paper 3, Section II, I

comment(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.

Paper 4, Section II, I

commentLet $H$ 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.

Paper 1, Section II, G

comment(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.]

Paper 2, Section II, G

comment(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.]

Paper 3, Section II, G

commentState 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 $\ell_{1}^{n} \rightarrow \mathrm{c}_{0}$.

Paper 4, Section II, G

commentLet $H$ 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 $T$.

Paper 1, Section II, G

commentLet $X$ 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.]

Paper 2, Section II, G

comment(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 $\left\|x_{i}\right\| \leqslant K\|x\|(i=1,2) .$

Paper 3, Section II, G

comment(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 .

Paper 4, Section II, G

commentDefine 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.]

Paper 1, Section II, F

commentState 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.

Paper 2, Section II, F

commentLet $X$ 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 $\ell_{\infty}$.

Paper 3, Section II, F

commentState 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 $C(K)$.

Paper 4, Section II, F

commentLet $T: X \rightarrow X$ 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.

Paper 1, Section II, G

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.

- $T f=f\left(\frac{1}{2}\right)$