Brent Nelson
Michigan State University, Department of Mathematics
Homework is collected at the beginning of lecture of the specified day. No late homework will be accepted. Please write your name and the homework number on each assignment. If you would like to try writing your homework with LaTeX, here is a template that produces this output.
Homework 7, due Friday, April 24th (Sections V.6-8,12) Solutions
1. Let $X$ be a Tychonoff space. Show that $X$ is open in $\beta X$ if and only if $X$ is locally compact.
2. For $(x_n)_{n\in \mathbb{N}} \in \ell^\infty(\mathbb{N})$, show that $a\in \mathbb{F}$ is a cluster point of the sequence if and only if there exists $\omega\in \beta\mathbb{N}\setminus\mathbb{N}$ with $\displaystyle \lim_{n\to\omega} x_n = a$.
3. Fix an enumeration $\{r_n\colon n\in \mathbb{N}\} = \mathbb{Q}\cap [0,1]$. For $s\in [0,1]\setminus \mathbb{Q}$, define $N_s:=\{n_k\colon k\in \mathbb{N}\}$ where $(r_{n_k})_{k\in \mathbb{N}}$ converges to $s$. Identifying $N_s\subset \beta\mathbb{N}$, define $A_s:= \overline{N_s}\setminus\mathbb{N}$.
(a) Show that for distinct $s,t\in [0,1]\setminus\mathbb{Q}$, $N_s\cap N_t$ is finite.
(b) Show that $\{A_s\colon s\in [0,1]\setminus\mathbb{Q}\}$ is a pairwise disjoint family of clopen subsets in $\beta\mathbb{N}\setminus\mathbb{N}$.
4. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space.
(a) For $1 < p < \infty$, show that $\text{ext}(L^p(X,\mu))_1 = \{f\in L^p(X,\mu)\colon \|f\|_p = 1\}.$
(b) Show that \[ \text{ext}(L^1(X,\mu))_1 = \{\alpha 1_E\colon E\text{ is an atom of $\mu$ and $\alpha\in \mathbb{F}$ with }|\alpha| = \mu(E)^{-1}\}. \]
(c) Show that $\text{ext}(L^\infty(X,\mu))_1 = \{f\in L^\infty(X,\mu)\colon |f(x)| = 1\text{ $\mu$-a.e.}\}$.
5. Show that $(\ell^1(\mathbb{N}))_1$ equals the norm closure of $\text{co}(\text{ext}(\ell^1(\mathbb{N}))_1)$.
6. Let $\mathcal{X}$ be a locally convex space, and let $K_1,\ldots, K_n\subset \mathcal{X}$ be compact convex subsets.
(a) Show that $\overline{\text{co}}(K_1\cup\cdots\cup K_n)=\text{co}(K_1\cup\cdots\cup K_n)$.
(b) Show that $\text{co}(K_1\cup\cdots \cup K_n)$ is compact.
7. Let $\mathcal{H}$ be a Hilbert space, and let $T\in \mathcal{B}(\mathcal{H})$ be such that either $T$ or $T^*$ is an isometry. Show that $T\in \text{ext}(\mathcal{B}(\mathcal{H}))_1$.
8. Let $\mathcal{A}\subset C_b(\mathbb{R})$ be the norm closed $*$-subalgebra generated by $\sin(t)$ and $\cos(t)$. Show that \[ \mathcal{A} = \{f\in C_b(\mathbb{R}) \colon f(t+2\pi)= f(t)\ \forall t\in \mathbb{R}\}. \]
9. For $m,n\in \mathbb{N}$ with $m < n$, define $x_{m,n}\in \ell^2(\mathbb{N})$ by \[ x_{m,n}(k) =\begin{cases} 1 &\text{if }k=m\\ m &\text{if }k=n\\ 0 &\text{otherwise} \end{cases}. \] Denote $S:=\{x_{m,n}\colon m,n\in \mathbb{N},\ m < n\}$.
(a) Show that the weak closure of $S$ contains $0\in \ell^2(\mathbb{N})$.
(b) Show that $0\in \ell^2(\mathbb{N})$ is not the limit of any sequence from $S$.
10. Let $\mathcal{X}$ be a Banach space. Show that there exists a Banach space $\mathcal{X}_*$ satisfying $(\mathcal{X}_*)^*=\mathcal{X}$ if and only if $\mathcal{X}$ admits a locally convex topology $\mathscr{T}$ under which $(\mathcal{X})_1$ is compact.
[Hint: use $X_*:=\{x^*\in \mathcal{X}^*\colon x^*\mid_{(\mathcal{X})_1}\text{ is $\mathscr{T}$-continuous}\}$.]
Exercises 2, 5, 7, and 9 were graded, and the rest were checked for completion. Homework 6, due Wednesday, April 8th (Sections IV.3, V.1-5) Solutions
1. Let $\mathcal{X}$ be a LCS whose topology is determined by a family of seminorms $\mathscr{P}$. Show that if $f\colon \mathcal{X}\to\mathbb{F}$ is a continuous linear functional, then there exists $p_1,\ldots, p_n\in \mathscr{P}$ and $\alpha_1,\ldots, \alpha_n>0$ such that \[ |f(x)| \leq \sum_{i=1}^n \alpha_i p_i(x) \qquad \forall x\in \mathcal{X}. \]
2. Let $\mathcal{X}$ be a TVS and let $p\colon \mathcal{X}\to\mathbb{R}$ be a sublinear functional. Define $G:=\{x\in \mathcal{X}\colon p(x)<1\}$ and for $x\in \mathcal{X}$ define \[ q(x):=\inf\{t>0\colon x\in t G\}. \] Show that $q(x) = \max\{ p(x),0\}$.
3. Let $\mathcal{X}$ be a TVS. For $\mathcal{Y}\subset\mathcal{X}$, show that the following statements are equivalent:
(i) $\mathcal{Y}-y$ is a hyperplane for all $y\in \mathcal{Y}$.
(ii) $\mathcal{Y}-y_0$ is a hyperplane for some $y_0\in \mathcal{Y}$.
(iii) There exists a non-zero linear functional $f\colon\mathcal{X}\to\mathbb{F}$ and $\alpha\in \mathcal{F}$ such that $\mathcal{Y}=f^{-1}(\alpha)$.
We say $\mathcal{Y}$ is an affine hyperplane if it satisfies any (hence all) of the above conditions.
4. Let $\mathcal{H}$ be a Hilbert space, and suppose $(h_i)_{i\in I} \subset \mathcal{H}$ converges weakly to some $h_0\in \mathcal{H}$. Show that $\|h_i - h_0\|\to 0$ if and only if $\|h_i\| \to \|h_0\|$.
5. Let $\mathcal{X}$ be a normed space. Suppose $(x_n)_{n\in\mathbb{N}}\subset \mathcal{X}$ converges weakly to some $x_0\in \mathcal{X}$. Show that there exists a sequence $(y_n)_{n\in \mathbb{N}}$ with $y_n\in \text{co}\{x_1,\ldots, x_n\}$ such that $\| x_0- y_n\|\to 0$.
6. Let $X$ be a Hausdorff space. A function $f\colon X\to\mathbb{R}$ is said to be \textbf{lower semi-continuous} if whenever a net $(x_i)_{i\in I}\subset X$ converges to some $x_0$ one has \[ f(x_0) \leq \liminf_{i\to\infty} f(x_i). \]
(a) Suppose $X$ is a compact Hausdorff space and $f\colon X\to\mathbb{R}$ is bounded below and lower semi-continuous. Show that $f$ achieves its minimum value.
(b) Let $\mathcal{X}$ be a normed space. Show that $\mathcal{X}\ni x\mapsto \|x\|$ is lower semi-continuous with respect to the weak topology.
(c) Let $\mathcal{X}$ be a normed space. Show that $\mathcal{X}^*\ni x^*\mapsto \|x^*\|$ is lower semi-continuous with respect to the weak* topology.
7. Let $\mathcal{X}$ be an infinite-dimensional normed space. Show that the weak closure of $\{x\in \mathcal{X}\colon \|x\| =1\}$ is $\{x\in \mathcal{X}\colon \|x\|\leq 1\}$. Then use this to argue that the norm topology and weak topology are distinct.
8. Let $\mathcal{X}$ be a LCS and let $V\subset \mathcal{X}$ be an open neighborhood of zero. Show that its polar \[ V^\diamond = \{x^*\in \mathcal{X}^*\colon |(x,x^*)|\leq 1,\ \forall x\in V\} \] is weak* compact in $\mathcal{X}^*$.
9. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space. A measurable subset $A\subset X$ is called an atom if $\mu(A)>0$ and for any measurable $E\subset A$ one has $\mu(E)=0$ or $\mu(E)=\mu(A)$.
(a) For $1\leq p \leq \infty$, show that $L^p(X,\mu)$ is finite dimensional if and only if $X$ can be partitioned into a finite union of atoms.
(b) Show that $L^1(X,\mu)$ is reflexive if and only if it is finite-dimensional.
(c) Show that $L^\infty(X,\mu)$ is reflexive if and only if it is finite-dimensional.
10. For $\mu,\nu\in M[0,1]$, define \[ d(\mu,\nu)=\sum_{n=0}^\infty 2^{-n} \left| \int_0^1 x^n\ d\mu(x) - \int_0^1 x^n\ d\nu(x) \right|. \]
(a) Show that $d$ is a metric on $M[0,1]$.
(b) Show that the topology defined by this metric is the same as the weak* topology on $(M[0,1])_1$.
(c) Show that the topology defined by this metric is not the same as the weak* topology on $M[0,1]$.
[Hint: construct a sequence $(\mu_n)_{n\in \mathbb{N}} \subset M[0,1]$ that converges to zero with respect to the metric, but does not converge to zero weak*.]
Exercises 1, 5, 7, and 9 were graded, and the rest were checked for completion. Homework 5, due Wednesday, March 25th (Sections III.10-III.14, IV.1-IV.2) Solutions
1. Let $X$ be a normed space with $Y\leq X$, let $\rho_X\colon X\to X^{**}$ and $\rho_Y\colon Y\to Y^{**}$ be the natural maps, and let $\iota\colon Y\to X$ be the inclusion map.
(a) Show that there exists an isometry $\phi\colon Y^{**}\to X^{**}$ such that $\phi\circ \rho_Y = \rho_X\circ \iota$.
(b) Prove that $\phi(Y^{**}) = (Y^\perp)^\perp$.
2. Let $X$ and $Y$ be normed spaces and let $A\colon X\to Y$ be a linear transformation. Show that $\text{graph}(A)$ is closed if and only if $x_n\to 0$ in $X$ and $Ax_n \to y$ in $Y$ implies $y=0$.
3. Let $X$ be a vector space with two norms $\|\, \cdot\, \|_1$ and $\|\, \cdot\, \|_2$, and assume $X$ is complete with respect to both norms. Denote by $\mathscr{T}_1$ and $\mathscr{T}_2$ the topologies induced by the norms and suppose $\mathscr{T}_1\supset \mathscr{T}_2$. Show that $\mathscr{T}_1=\mathscr{T}_2$.
4. Let $X$ and $Y$ be Banach spaces and $A\in \mathcal{B}(X,Y)$. Show that there exists $c>0$ such that $\|A x\|\geq c \|x\|$ for all $x\in X$ if and only if $\ker(A)=\{0\}$ and $\text{ran}(A)$ is closed.
5. Let $X$ be a Banach space. Show that if $Y\leq X$ is complemented, then all subspaces complementary to $Y$ are isomorphic to $X/Y$.
6. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space. For $(f_n)_{n\in \mathbb{N}}\subset L^1(X,\mu)$, show that $\int_X f_n g\ d \mu\to 0$ for all $g\in L^\infty(X,\mu)$ if and only if $\sup \|f_n\|_1<\infty$ and $\int_E f_n\ d\mu\to 0$ for all $E\in \Omega$.
7. Let $(S,d)$ be a metric space and let $X$ be a normed space. A map $f\colon S\to X$ is said to be Lipschitz if there exists a constant $C>0$ so that $\|f(s) - f(t)\| \leq C d(s,t)$ for all $s,t\in S$. Show that $f\colon S\to X$ is Lipschitz if and only if $x^*\circ f\colon S\to \mathbb{C}$ is Lipschitz for all $x^*\in X^*$.
8. Let $X$ be a vector space and let $\mathscr{P}$ be a collection of seminorms on $X$. Let $\mathscr{T}$ be the topology induced by $\mathscr{P}$; that is, $\mathscr{T}$ is the smallest collection of sets closed under unions and finite intersections containing all sets of the form $U_{p,x_0,\epsilon}:=\{x\in X\colon p(x-x_0)<\epsilon\}$ for $p\in\mathscr{P}$, $x_0\in X$, and $\epsilon>0$.
(a) Show that every element of $\mathscr{T}$ is of the form $\bigcup_{i\in I} F_i$, where for each $i\in I$, $F_i=U_{p_1,x_1,\epsilon_1}\cap\cdots \cap U_{p_n,x_n,\epsilon_n}$ with $p_1,\ldots, p_n\in \mathscr{P}$, $x_1,\ldots, x_n\in X$, and $\epsilon_1,\ldots, \epsilon_n>0$.
(b) Show that $U\in \mathscr{T}$ if and only if for every $x_0\in U$ there exists $p_1,\ldots, p_n\in \mathscr{P}$, $x_1,\ldots, x_n\in X$, and $\epsilon_1,\ldots, \epsilon_n>0$ such that $x_0\in U_{p_1,x_1,\epsilon_1}\cap\cdots \cap U_{p_n,x_n,\epsilon_n}\subset U$.
(c) Show that a net $(x_i)_{i\in I}\subset X$ converges to $x_0\in X$ in the $\mathscr{T}$-topology if and only if $\lim p(x_i-x_0)=0$ for all $p\in \mathscr{P}$.
9. Let $X$ and $Y$ be locally convex spaces and let $T\colon X\to Y$ be a linear transformation. Show that $T$ is continuous if and only if for every continuous seminorm $p$ on $Y$, $p\circ T$ is a continuous seminorm on $X$.
10. Let $X$ be a vector space and let $(p_n)_{n\in \mathbb{N}}$ be a sequence of seminorms on $X$ satisfying \[ \bigcap_{n=1}^\infty \ker(p_n) = \{0\}. \]
(a) Show that \[ d(x,y):= \sum_{n=1}^\infty 2^{-n} \frac{p_n(x-y)}{1+p_n(x-y)},\qquad x,y\in X \] defines a metric on $X$.
(b) Show that the topology induced by the metric $d$ is equal to the topology induced by the family of seminorms $\{p_n\colon n\in\mathbb{N}\}$.
Exercises 1, 3, 7, and 10 were graded, and the rest were checked for completion. Homework 4, due Friday, February 28th (Sections II.8, III.1-III.6) Solutions
1. Let $P,Q\in \mathcal{B}(\mathcal{H})$ be projections. Show that $P\cong Q$ if and only if $\dim(\text{ran}\,P)=\dim(\text{ran}\,Q)$ and $\dim(\ker{P})=\dim(\ker{Q})$.
2. Let $X$ be a locally compact Hausdorff space. For $f\colon X\to\mathbb{F}$, the support of $f$ is the set \[ \text{supp}(f):=\overline{\{x\in X\colon f(x)\neq 0\}}. \] Let $C_c(X)$ denote the set of continuous functions with compact support, which we note are contained in $C_0(X)$. Show that $C_c(X)$ is dense in $C_0(X)$.
3. Let $X$ and $Y$ be compact Hausdorff spaces, and let $\phi\colon Y\to X$ be a continuous map. Define $L\in \mathcal{B}(C(X),C(Y))$ by $L(f)=f\circ \phi$.
(a) Find necessary and sufficient conditions on $\phi$ which make $L$ injective.
(b) Find necessary and sufficient conditions on $\phi$ which make $L$ surjective.
(c) Find necessary and sufficient conditions on $\phi$ which make $L$ isometric.
(d) Suppose $X=Y$. Show that $L^2=L$ if and only if $\phi\circ\phi=\phi$.
4. Show that a locally compact normed space $\mathcal{X}$ is necessarily finite dimensional.
5. Let $X$ be a locally compact normal space, let $V\subset X$ be a closed subset, and let \[ \mathcal{Y}:=\{f\in C_0(X)\colon V\subset \ker(f)\}. \]
(a) Show that $\mathcal{Y}\leq C_0(X)$.
(b) Show that $C_0(X)/\mathcal{Y}$ is isometrically isomorphic to $C_0(V)$.
6. Let $\{\mathcal{X}_i\colon i\in I\}$ be a collection of normed spaces. For $1\leq p<\infty$, show that $\left(\bigoplus_{i\in I}^p \mathcal{X}_i\right)^*$ is isometrically isomorphic to $\bigoplus_{i\in I}^q \mathcal{X}_i^*$ where $\frac1p + \frac1q = 1$.
7. Let $C^n [0,1]$ denote the functions $f\colon [0,1]\to\mathbb{F}$ with $n$ continuous derivatives, which we equip with the following norm: \[ \|f\| := \max_{0\leq k\leq n} \|f^{(k)}\|_\infty. \]
(a) Show that the above norm is equivalent to the following norm \[ \|f\|_s:= \sum_{k=0}^{n-1} |f^{(k)}(0)| + \|f^{(n)}\|_\infty. \]
(b) For $L\in C^n[0,1]^*$, show that there exists $\alpha_0,\ldots, \alpha_{n-1}\in\mathbb{F}$ and a measure $\mu$ on $[0,1]$ such that \[ L(F) = \sum_{k=0}^{n-1} \alpha_k f^{(k)}(0) + \int_0^1 f^{(n)}\ d\mu. \] Also, compute $\|L\|$ in terms of $|\alpha_0|,\ldots,|\alpha_{n-1}|$ and $\|\mu\|$.
8. For $n\in \mathbb{N}$, let $\mathbb{P}_n$ denote the polynomials of degree at most $n$.
(a) Show that for fixed $n\in \mathbb{N}$ there exists $\mu\in M([0,1])$ such that $\int_0^1 p\ d\mu=p'(0)$ for all $p\in \mathbb{P}_n$.
(b) Show that there is no $\mu\in M([0,1])$ such that $\int_0^1 p\ d\mu=p'(0)$ for all $p\in \mathbb{P}_n$ and all $n\in \mathbb{N}$.
9. Let $c(\mathbb{N})$ denote the sequences $(x_n)_{n\in \mathbb{N}}\subset \mathbb{F}$ such that $\lim_n x_n$ exists.
(a) Show that $c(\mathbb{N})\leq \ell^\infty(\mathbb{N})$.
(b) Prove that $\ell^\infty(\mathbb{N})^*\not\cong \ell^1(\mathbb{N})$.
10. Let $\mathcal{X}$ be a Banach space. Show that if $\mathcal{X}^*$ is separable then $\mathcal{X}$ is separable.
[Hint: Let $\{f_n\colon n\in \mathbb{N}\}$ be a countable dense subset of $\mathcal{X}^*$, and for each $n\in\mathbb{N}$ let $x_n\in \mathcal{X}$ be such that $|f_n(x_n)|\geq \frac12 \|f_n\|$. Show that $\overline{\text{span}}\{x_n\colon n\in \mathbb{N}\}=\mathcal{X}$.]
Exercises 2, 4, 9, and 10 were graded, and the rest were checked for completion. Homework 3, due Monday, February 17th (Sections II.3 - II.5, II.7) Solutions
1. Let $\{\mathcal{K}_i\colon i\in I\}$ be a collection of closed subspaces of a Hilbert space $\mathcal{H}$. Prove that \[ \left(\bigcup_{i\in } \mathcal{K}_i \right)^\perp = \bigcap_{i\in } \mathcal{K}_i^\perp \qquad \text{ and }\qquad \left(\bigcap_{i\in I} \mathcal{K}_i \right)^\perp = \overline{\text{span}} \bigcup_{i\in I} \mathcal{K}_i^\perp. \]
2. Let $P,Q\in \mathcal{B}(\mathcal{H})$ be projections.
(a) Show that $P+Q$ is a projection if and only if $P$ and $Q$ are orthogonal.
(b) If $P+Q$ is a projection, show that $\text{ran}(P+Q)=\text{ran}(P)+\text{ran}(Q)$ and $\ker(P+Q)=\ker(P)\cap \ker(Q)$.
(c) Show that $PQ$ is a projection if and only if $P$ and $Q$ commute.
(d) If $PQ$ is a projection, show that $\text{ran}(PQ)=\text{ran}(P)\cap\text{ran}(Q)$ and $\text{ker}(PQ)=\ker(P)+\ker(Q)$.
3. Let $E\in\mathcal{B}(\mathcal{H})$ be an idempotent. Show that $E\in K(\mathcal{H})$ if and only if $E\in \mathcal{FR}(\mathcal{H})$.
4. Let $T\in \mathcal{FR}(\mathcal{H})$.
(a) Show that there exists an orthonormal set $\{e_1,\ldots, e_n\}\subset \mathcal{H}$ and non-zero vectors $f_1,\ldots, f_n\in \mathcal{H}$ so that \[ Th = \sum_{i=1}^n \left\langle h,e_i \right\rangle f_i \qquad\forall h\in \mathcal{H}. \]
(b) Prove that $\dim\text{ran}(T) = \dim\text{ran}(T^*)$.
(c) Show that $T$ is normal if $f_i=\lambda_i e_i$ for some scalar $\lambda_i$, for each $i=1,\ldots, n$. Show that the converse holds when $\mathcal{H}$ is a complex Hilbert space. Determine $\sigma_p(T)$ in this case.
(d) Show that $T$ is a projection if and only if $f_i = e_i$ for each $i=1,\ldots, n$.
5. Let $\{\mathcal{H}_n\colon n\in \mathbb{N}\}$ be a family of Hilbert spaces, and let $T_n\in \mathcal{B}(\mathcal{H}_n)$ be a family of bounded operators satisfying $\sup_n \|T_n\|<\infty$. Define $\mathcal{H}:= \bigoplus_n \mathcal{H}_n$ and $T\in \mathcal{B}(\mathcal{H})$ by \[ T(h_n)_{n\in \mathbb{N}} = (T_n h_n)_{n\in \mathbb{N}}. \] Show that $T$ is compact if and only if each $T_n$ is compact and $\lim_n \|T_n\|=0$.
6. Show that for non-zero $\phi\in L^\infty([0,1],m)$, $M_\phi$ is not compact.
7. Let $V$ be the Volterra operator: for $f\in L^2([0,1],m)$ \[ (Vf)(x)=\int_0^x f(y)\ dy. \] In this exercise you will show that $\sigma_p(V)=\emptyset$. Suppose $f\in \ker(V-\lambda)$ for some scalar $\lambda$.
(a) Show that $Vf=0$ implies $f=0$.
(b) For $\lambda\neq 0$, show that $f$ is continuous by proving \[ |f(x) - f(y)| \leq \frac{\|f\|_2}{|\lambda|} |x-y|^{1/2}\qquad x,y\in [0,1]. \]
(c) For $\lambda\neq 0$, show that $f$ is differentiable on $(0,1)$ and satisfies $f'=\frac{1}{\lambda} f$.
(d) Conclude that $f=0$ and $\sigma_p(V)=\emptyset$.
8. Let $\{P_i\colon i\in I\}$ be a family of pairwise orthogonal projections in $\mathcal{B}(\mathcal{H})$.
(a) Show that for each $h\in \mathcal{H}$ \[ \sum_{i\in I} P_i h \] converges and is equal to $P_{\mathcal{K}}h$ where \[ \mathcal{K}=\bigoplus_{i\in I} \text{ran}(P_i). \]
(b) Show that if $I$ is infinite, then for any $F\subset I$ finite \[ \left\| P_\mathcal{K} - \sum_{i\in F} P_i \right\| = 1. \] Conclude that $\sum_{i\in I} P_i$ does not converge to $P_\mathcal{K}$ with respect to the operator norm.
9. Let $\mathcal{H}$ be a Hilbert space.
(a) For $f,g\in \mathcal{H}$, define $f\otimes\bar{g}\in \mathcal{FR}(\mathcal{H})$ by $(f\otimes\bar{g})(h) = \langle h,g \rangle f$. Show that $T$ commutes with $f\otimes \bar{g}$ if and only if $f\in \ker(T-\lambda)$ and $g\in \ker(T^*-\bar\lambda)$ for some $\lambda\in \mathbb{C}$.
(b) Show that $\mathcal{FR}(\mathcal{H})' = \mathbb{C}$. Deduce that $K(\mathcal{H})'=\mathbb{C}$ and $\mathcal{B}(\mathcal{H})' = \mathbb{C}$.
(c) Let $P\in \mathcal{B}(\mathcal{H})$ be a projection. Show that $\{P\}'' = \mathbb{C}P + \mathbb{C}(1-P)$.
10. Let $T\in K(\mathcal{H})$ be normal. Show that $\dim\ker(T-\lambda)\leq 1$ for all $\lambda\in \mathbb{C}$ if and only if there exists $h\in \mathcal{H}$ such that $\{p(T)h\colon p\text{ a polynomial}\}$ is dense. (In this case $h$ is called a cyclic vector for $T$.)
Homework 2, due Friday, January 31st (Sections I.4 - II.2) Solutions
1. Let $\mathcal{H}$ be a Hilbert space. Show that the closed unit ball $\{h\in \mathcal{H}\colon \|h\|\leq 1\}$ is compact if and only if $\dim\mathcal{H} < \infty$.
2. Define $V\colon L^2(\mathbb{R},m) \to L^2(\mathbb{R},m)$ by \[ (Vf)(t) = f(t-1). \]
(a) Show that $V$ is an isomorphism.
(b) Consider $W\colon L^2((0,\infty),m)\to L^2((0,\infty),m)$ given by \[ Wf = 1_{(1,\infty)} Vf. \] Show that $W$ is an non-surjective isometry.
3. Let $\mathcal{H}$ be the Hilbert space from Homework 1, Exercise 1. Define $U\colon \mathcal{H}\to L^2([0,1],m)$ by $Uf=f'$. Show that $U$ is an isomorphism and find a formula for $U^{-1}$.
4. Let $\mu,\nu$ be two $\sigma$-finite measures on a measure space $(X,\Omega)$. Suppose $\nu\ll\mu$ and $\phi:=\frac{d\nu}{d\mu}$ is the Radon-Nikodym derivative. For $f\in L^2(X,\Omega,\nu)$ define $Vf := \sqrt{\phi} f$.
(a) Show that $V\colon L^2(X,\Omega,\nu)\to L^2(X,\Omega,\mu)$ and that $V$ is an isometry.
(b) Show that $V$ is an isomorphism if and only if $\mu\ll\nu$.
5. Prove that the map \[ (h_n)_{n\in \mathbb{N}}\mapsto \sum_{n\in \mathbb{N}} e_n\otimes h_n \] defines an isomorphism $\ell^2(\mathbb{N},\mathcal{H}) \cong \ell^2(\mathbb{N})\otimes \mathcal{H}$.
6. Let $\mathcal{H}$ be a Hilbert space with countable orthonormal basis $\mathcal{E}$. Suppose $A\colon \mathcal{E}\to \mathcal{H}$ satisfies \[ \sum_{e\in \mathcal{E}} \| Ae\| < \infty. \] Show that there exists a unique $B\in\mathcal{B}(\mathcal{H})$ such that $B\mid_\mathcal{E} = A$.
7. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space. For $\phi\in L^\infty(X,\Omega,\mu)$, show that $M_\phi^2 = M_\phi$ if and only if $\phi=1_S$ for some measurable subset $S\subset X$.
8. Let $\mathcal{H}$ be a Hilbert space. We say $A\in \mathcal{B}(\mathcal{H})$ is diagonalizable if there exists an orthonormal basis $\mathcal{E}$ and scalars $\{\alpha_e\colon e\in \mathcal{E}\}\subset\mathbb{F}$ such that $Ae= \alpha_e e$ for all $e\in \mathcal{E}$. Given an orthonormal basis $\mathcal{E}$ and scalars $\{\alpha_e\colon e\in \mathcal{E}\}$, show that there exists diagonalizable $A\in \mathcal{B}(\mathcal{H})$ with $Ae=\alpha_e e $ for all $e\in \mathcal{E}$ if and only if $\sup_e |\alpha_e| < \infty$. Also, show that in this case one has $\|A\| = \sup_e |\alpha_e|$.
9. Let $f(z)=\sum_{n=0}^\infty \alpha_n z^n$ be power series with radius of convergence $0< R \leq \infty$. Let $\mathcal{H}$ be a Hilbert space.
(a) For $A\in \mathcal{B}(\mathcal{H})$ with $\|A\| < R$, show that there exists $T\in \mathcal{B}(\mathcal{H})$ such that for any $h,g\in \mathcal{H}$ \[ \left\langle Th, g \right\rangle = \sum_{n=0}^\infty \alpha_n \left\langle A^n h, g \right\rangle. \] (We denote $f(A):=T$.)
(b) Show that \[ \lim_{N\to\infty} \left\| T - \sum_{n=0}^N \alpha _n A^n \right\| =0. \]
(c) Show that if $B\in \mathcal{B}(\mathcal{H})$ satisfies $AB=BA$, then $TB=BT$.
(d) Let $f(z) = e^z = \sum_{n=0}^\infty \frac{1}{n!} z^n$. Show that for any self-adjoint $A\in \mathcal{B}(\mathcal{H})$, $f(iA)$ is unitary.
10. Let $\mathcal{H}$ be a Hilbert space.
(a) Suppose $A\in \mathcal{B}(\mathcal{H})$ is normal. Show that $A$ is injective if and only if it has dense range.
(b) Find an example of a (non-normal) operator $B$ which is injective but does not have dense range.
(c) Find an example of a (non-normal) operator $C$ which is surjective but is not injective.
Exercises 1, 4, 7, and 9 were graded, and the rest were checked for completion. Homework 1, due Friday, January 17th (Sections I.1 - I.3) Solutions
1. Let $\mathcal{H}$ be the vector space over $\mathbb{F}$ consisting of all absolutely continuous functions $f\colon [0,1]\to \mathbb{F}$ such that $f(0)=0$ and $f'\in L^2([0,1],m)$. For $f,g\in \mathcal{H}$, define \[ \left\langle f,g\right\rangle:= \int_0^1 f'(t) \overline{g'(t)}\ dt. \]
(a) Prove that the above is an inner product and that $\mathcal{H}$ is a Hilbert space when equipped with it.
(b) For $0< t\leq 1$, define a linear functional $L\colon \mathcal{H}\to\mathbb{F}$ by $L(f)= f(t)$. Show that $L$ is bounded and compute $\|L\|$.
[Hint: find $g\in \mathcal{H}$ so that $L(f)=\left\langle f,g\right\rangle$.]
2. Let $\mathcal{H}$ be a Hilbert space over $\mathbb{R}$. Show that there is Hilbert space $\mathcal{H}_\mathbb{C}$ over $\mathbb{C}$ and a linear map $U\colon \mathcal{H}\to\mathcal{H}_\mathbb{C}$ satisfying:
(i) $\left\langle Uf,Ug\right\rangle=\left\langle f,g,\right\rangle$ for all $f,g\in \mathcal{H}$; and
(ii) for each $h\in \mathcal{H}_\mathbb{C}$ there are unique $f,g\in \mathcal{H}$ such that $h=Uf + i Ug$.
$\mathcal{H}_\mathbb{C}$ is called the complexification of $\mathcal{H}$.
3. Let $V$ be a vector space over $\mathbb{F}$. A degenerate inner product is a map $u\colon V\times V\to\mathbb{F}$ satisfying the same properties as an inner product except instead of being positive definite it is positive semi-definite: $u(x,x) \geq 0$ for all $x\in V$. In particular, $u(x,x)=0$ does not imply $x=0$.
(a) Convince yourself that the same proofs of the Cauchy--Schwarz and triangle inequalities still work for degenerate inner products (you do not need to turn this part in).
(b) Prove that $N:=\{x\in V\colon u(x,x)=0\}$ is a subspace of $V$.
(c) Show that the following map defined on the quotient space $V/N$ is an inner product: \[ \left\langle x+N, y+N\right\rangle := u(x,y), \]
4. Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{K}\leq \mathcal{H}$.
(a) Show that $I_\mathcal{H}-P_{\mathcal{K}} = P_{\mathcal{K}^\perp}$.
(b) Show that $P_{\mathcal{K}} P_{\mathcal{K}^\perp} =P_{\mathcal{K}^\perp} P_{\mathcal{K}}=0$.
(c) Note that $0=P_{\{0\}} = P_{\mathcal{K}\cap \mathcal{K}^\perp}$. Find a counterexample to $P_{\mathcal{K}_1} P_{\mathcal{K}_2} = P_{\mathcal{K}_1\cap \mathcal{K}_2}$ for $\mathcal{K}_1,\mathcal{K}_2\leq \mathcal{H}$.
5. Let $\mathcal{H}$ be a Hilbert space and let $A\subset \mathcal{H}$ be a subset.
(a) Show that $\overline{span}\, A$ equals the intersection of all closed subspaces $\mathcal{K}\leq \mathcal{H}$ satisfying $\mathcal{K} \supset A$.
(b) Show that $(A^\perp)^\perp =\overline{span}\,A$.
6. Let $C^1$ be the set of continuous functions on $[0,1]$ with continuous derivative. For $t\in (0,1)$ define a linear functional $L\colon C^1\to \mathbb{F}$ by $L(f)=f'(t)$. Show that $L$ is not the restriction of a bounded linear functional on $L^2([0,1],m)$.
7. Show that every finite-dimensional vector space admits an inner product making it into a Hilbert space.
Exercises 1, 4, 5, and 7 were graded, and the rest were checked for completion.
Office Hours (D215 Wells Hall):
Wednesdays 2:30 - 3:30 pm
Course Notes:
I. Hilbert Spaces
II. Operators on Hilbert Spaces
III. Banach Spaces
IV. Locally Convex Spaces
V. Weak Topologies
VII. Banach Algebras