Announcements:

• Our first class meeting is Monday (the 31st) at 2:10
• If you are an undergraduate and would like to enroll in the course, please let me know.
• As of now, we will not hold class on Friday, April 11th; Wednesday, April 23; and Wednesday, May 7th. We'll make these up in our Thrusday x-hour on April 10th, April 24th, and May 15th.

Homework Assigments

Week of April 7 to 11
Assignments Made on:

Monday: Work: Since we are just reviewing compactness for metric spaces, I have put Jody Trout's handout from this past Fall's Math 73/103 in the Files folder on our canvas page for easy reference. Show that $X$ is compact if and only if given any family $\mathcal F$ of closed sets in $X$ with the finite intersection property we have $\bigcap_{F\in\mathcal F}F\not=\emptyset$. OP: Show that $E\subset X$ is totally bounded if and only if there is an $\epsilon$-net for $E$ for all $\epsilon >0$. OP: Suppose that $(X,\rho)$ is compact and that $f:(X,\rho)\to (Y,\sigma)$ is continuous. Show that $f(X)$ is compact in $Y$. Let $X$ be a metric space. Show that if $E$ is a compact subpace of $X$, then $E$ is closed. Show that if $X$ is compact and $E$ is closed in $X$, then $E$ is compact.

Wednesday: Work: Recall that there is no lecture Friday and that we are meeting in our x-hour tomorrow (Thursday) from 1:20 to 2:10 in our usual room. We say that $D$ is dense in $X$ if $\overline D=X$. Show that $D$ is dense if and only if $D$ meets every nonempty open set in $X$. OP: A metric space is separable if it has a countable dense subset. Show that a metric space is separable if and only if there is a countable family $\mathcal D$ of open sets such that every open set in $X$ is a union of elements from $D$: for all $U$ open in $X$, $U=\bigcup\{\, V:\text{$V\in\mathcal D$ and $V\subset U$}\,\}$. (Recall that a countable union of countable sets is countable.) OP: Let $X=(0,1)$. For each $x\in X$, let $B_{\delta_x}(x)=\{\,y\in (0,1):|x-y|< \delta\,\}$ be such that $y\in B_{\delta_{x}}(x)$ implies $\bigl|\frac1x-\frac1y\bigr|<1$. Show that the cover $$(0,1)=\bigcup_{x\in(0,1)} B_{\delta_x}(x)$$ has no Lebesgue number. Show that a compact metric space has a countable dense subset. (Actually, it is enough for the space to be totally bounded.)

Thursday (x-hour): Work: OP: Let $\mathcal F$ be the family of functions $f_n(x)=x^n$ on $X=[0,1]$. Show that $\mathcal F$ is equicontinuous at each $x\in [0,1)$. (Luke, invoke the force in the form of the Mean Value Theorem.) Show that an equicontinuous family of functions on a compact metric space is uniformly equicontinuous as stated in lecture. (Some texts do not define equicontinuous at a point. Instead, whether $X$ is compact or not, equicontinuity is what we have called uniformly equicontinuity. Fortunately, there is no distinction for compact spaces.) Show that if $X$ a metric space which is not totally bounded, then there is an unbounded continuous function $f:X\to\mathbf R$. I suggest the following. There is a $r>0$ and $\{x_n\}\subset X$ such that the $r$-balls $\{ B_r(x_n)\}$ are pairwise disjoint. That is, if $n\not= m$, then $B_r(x_n)\cap B_r(x_m)=\emptyset$. Show that there is a continuous function $f_n:X\to[0,1]$ such that $f_n(x_n)=1$ and $f_n(x)=0$ if $x\notin B_{\frac r2}(x_n)$. Consider $\sum n f_n$. Let $X$ be a metric space such that every continuous function $f:X\to\mathbf R$ attains its minimum value. Show that $X$ is complete. I suggest the following. Let $(x_n)$ be a Cauchy sequence in $X$. If $x\in X$, show that $(\rho(x,x_n))$ is Cauchy in $\mathbf R$. If $f(x)=\lim_n \rho(x,x_n)$, then show that $f$ is continuous on $X$. Conclude that there is $x_0\in X$ such that $f(x_0)=0$. Hence $x_n \to x_0$ and $X$ is complete. Show that a metric space is compact if and only if every continuous real-valued function on $X$ attains its maximum. (Note that every real-valued function attains it maximum if and only if every real-valued function attains its minimum. Consider $-f$.)

Friday: Work: No Lecture Today

Week of April 14 to 18
Assignments Made on:

Monday: Work: Show that $X$ is a Baire space if and only if whenever a countable union $\bigcup F_n$ of closed sets in $X$ has interior in $X$ at least one of the sets $F_n$ has interior in $X$. (In this problem, we will assume that if $(X,\rho)$ and $(Y,\sigma)$ are metric spaces then so is $(X\times Y,\delta)$ where $\delta((x,y),(x',y'))=\rho(x,x')+\sigma(y,y')$. You can also assume that with respect to this product metric, $(x_n,y_n)\to (x,y)$ if and only if $x_n\to x$ and $y_n\to y$. In particular, if $(X,\rho)$ and $(Y,\sigma)$ are complete, so is $(X\times Y,\delta)$.) Let $U$ be a nonempty open subset of a complete metric space $(X,\rho)$. Show that $U$ admits a complete metric which is equivalent to that inherited from $X$. I suggest the following. It suffices to find a homeomorphism $\phi:(U,\rho)\to (Y,\sigma)$ where $(Y,\sigma)$ is complete. Let $A=X\setminus U$ and define $f:U\to \mathbf R$ by $f(x)=\rho(x,A)^{-1}$. Then the map $\phi(x)=(x,f(x))$ is continuous from $(U,\rho)$ to $(X\times\mathbf R,\delta)$ where $\delta$ is the obvious complete product metric. It suffices to see that that the range of $\phi$ is closed. OP: The ruler function is an example of a function $f:\mathbf R\to \mathbf R$ which continuous at every irrational and discontinuous at each rational. In this problem, we want to see that it is impossible to construct a function which is continous exactly on the rationals. In fact, we are to prove that if $D$ is a countable dense subset of $\mathbf R$, then there is no function $f:\mathbf R\to \mathbf R$ such that the set of points $C$ where $f$ is continuous is equal to $D$. I suggest the following. Let $U_n$ be the union of all open sets $U\subset \mathbf R$ such that $\operatorname{diam}(f(U))<\frac1n$. Show that $C=\bigcap_n U_n$. (A subset of $\mathbf R$, such as $C$, which is the countable intersection of open sets is called a $G_\delta$ subset). Show that $D$ can't be a $G_\delta$ subset. (Consider: if $D=\bigcap W_n$ and $V_d:=\mathbf R\setminus \{d\}$ for each $d\in D$, then $W_n$ and $V_d$ are dense open subsets of $\mathbf R$. OP: Every vector space $V$ has a basis --- that is, a linearly independent subset $B$ such that every element in $V$ is a finite linear combination of elements of $B$. The dimension of $V$, $\operatorname{dim} V$, is the cardinality of any such basis. (In analysis, such a basis is sometimes called a Hamel basis to stress that it is a bonifide vector space basis.) Show that if $V$ is a Banach space, then its dimension is either finite or uncountable. (Use problem #26.)

Wednesday: Turn in: Please turn in problems 13, 16, 17, 20, 22, 23, 24, 25, 26, and 27 by lecture on Monday the 21st. Let me know if you need more time. Work: Suppose that $X$ and $Y$ are normed vector spaces. Show that $\mathcal L(X,Y)$ is a normed vector space with respect to the "operator norm" defined in lecture such that \[ \|T(x)\|\le \|T\|\|x\|. \] Also show that if $S\in \mathcal L(Y,Z)$ then $\|ST\|\le \|S\|\|T\|$. Show that \[ \|T\|=\inf\{ \alpha\ge0: \|T(x)\|\le \alpha\|x\|\quad\text{for all $x\in X$}\}.\] Suppose $X$ and $Y$ are Banach spaces with $T\in \mathcal L(X,Y)$. Suppose that $E$ is a closed proper subspace of $X$ such that $E\subset \ker T$. Show that there is a unique operator $\overline{T}\in\mathcal L(X/E,Y)$ such that $\overline{T}(q(x))=T(x)$ for all $x\in X$ where $q:X\to X/E$ is the quotient map. Moreover, $\|\overline T\|=\|T\|$. OP: Suppose that $X$ and $Y$ are Banach spaces, that $D$ is a dense subspace of $X$ and that $T_0\in\mathcal L(D,Y)$. Show that there is a unique $T\in \mathcal L(X,Y)$ such that $T(x)=T_0(x)$ for all $x\in D$. (Let $(x_n)$ and $(y_n)$ be sequences in $D$ converging to $x\in X$. Show that $(T(x_n))$ and $T(y_n))$ must converge to the same element of $y$.) For Fun Only: The existence of continuous functions that fail to have a derivative at any point (aka nowhere differentiable) was greeted with sckepticism when Wierestrass first proved such things existed. He was forced to produce an example. (Spivak produces a simpler version of Wierestrass's example in his Calculus book (see Chapter 23, Theorem 5).) Using the Baire Category Theorem, we can easily see that the set of continuous nowhere differentiable functions is dense in $C[0,1]$. My proof of this is attached for your amusement.

Friday: Work: Let $E$ and $X$ be Banach spaces with $E$ finite dimensional. Show that every linear map $S:E\to X$ is bounded. Show that a linear map $T:X\to E$ is bounded if and only if $\ker T$ is closed. Suppose that $E$ and $M$ are closed subspaces of a Banach space $X$. If $E$ is finite dimensional, show that $E+M=\{\, x+y : \text{$x\in E$ and $y \in M$}\,\}$ is closed.

Week of April 21 to 25
Assignments Made on:

Monday: Work: Suppose that $X$ and $Y$ are Banach spaces and $T\in\mathcal L(X,Y)$. Show that $T$ is injective with closed range if and only if $$\inf\{\, \|T(x)\|: \|x\|=1\,\}>0.$$ The next two (optional) problems demonstrate that the Open Mapping Theorem requires both the domain $X$ and the range $Y$ to be Banach spaces. The conclusion does not follow if only one is assumed to be complete. OP: Let $(X,\|\cdot\|)$ be an infinite-dimensional Banach space and $f:X\to\mathbf F$ a discontinuous linear functional on $X$. Show that $\|x\|_0 := \|x\|+|f(x)|$ is a norm on $X$ and that the identity map $\operatorname{id}: (X,\|\cdot\|_0)\to (X,\|\cdot\|)$ is a surjective bounded linear map that is not an open map. OP: Let $C_0^1([0,1])$ be the subspace of the Banach space $(C([0,1]),\|\cdot\|_\infty)$ consisting of functions $f$ such that $f(0)=0$ and such that $f^\prime$ is continuous on $[0,1]$. Define $\Phi:C([0,1]) \to C_0^1([0,1])$ by $\Phi(f)(x)=\int_0^xf(t)\,dt$. Show that $\Phi$ is a bounded surjection that is not an open map.

Wednesday: Work: No Lecture Today

Thursday (x-hour): Work: Let $X$ be a normed vector space. A Banach space $\widetilde X$ is called a completion of $X$ is there is an isometic isomorphism $\iota:X\to \widetilde X$ onto a dense subspace of $\widetilde X$. Show that any two completions $(\widetilde X_1,\iota_1)$ and $(\widetilde X_2,\iota_2)$ are isometrically isomorphic by an isomorphism $\Phi:\widetilde X_1\to \widetilde X_2$ such that $\Phi(\iota_1(x))=\iota_2(x)$. (An isomorphism is isometric if it is norm preserving. This exercise allows us to abuse language slightly and talk about the completion of $X$.)

Friday: Work: Let's find a use for a genuine Minkowski functional. In this problem, we'll let $\ell^\infty_{\mathbf R}$ be the real Banach space of bounded sequences in $\mathbf R$. Define $m$ on $\ell^\infty_{\mathbf R}$ by $m(x)=\limsup_n x_n$. We clearly have $m( t x)=tm(x)$ if $t\ge 0$ and it is not hard to check that $m(x+y)\le m(x)+m(y)$. (You may take this as given.) We want to show that there are Banach limits or what I prefer to call a generalized limit on $\ell^\infty_{\mathbf R}$. That is we want to show that there is a functional $L\in {\ell^\infty_{\mathbf R}}^*$ such that $L(S(x))=L(x)$ where $S\in \mathcal L(\ell^\infty_{\mathbf R})$ is given by $S(x)_n=x_{n+1}$ and such that $\liminf_n x_n \le L(x) \le \limsup_n x_n$. In particular, $L(x)=\lim_n x_n$ wherever the limit exists. Here's what I suggest. Define $$m_n(x) =\frac1n(x_1+\cdots +x_n).$$ Let $Y$ be the subspace of $\ell^\infty_{\mathbf R}$ for which $\lim_n m_n(x)$ exists and define $L_0$ on $Y$ by $L_0(x)=\lim_n m_n(x)$. Now use the Basic Extension Lemma to extend $L_0$ to $\ell^\infty_{\mathbf R}$. Note that $x-S(x)$ is in $Y$. Prove the following Lemma from lecture: Let $X$ be a complex vector space. Every real linear functional of $X$ is the real part of a unique complex linear functional on $X$. In fact, if $\phi=\operatorname{Re}(\psi)$ then $\psi(x)=\phi(x)-i\phi(ix)$.

Week of April 28 to May 2
Assignments Made on:

Monday: Work: Suppose that $X$ is a normed vector space such that $X^*$ is separable. Show that $X$ is separable. (Hint: Let $(\phi_n)$ be a dense sequence in $X^*$. Find $(x_n)$ in $X$ such that each $\|x_n\|=1$ and $|\phi_n(x_n)|\ge \frac12\|\phi_n\|$. Then show that the space $S$ spanned by $\{\,x_n:n\in \mathbf N\,\}$ is dense by considering an appropriate linear functional annihilating $S$.)

Wednesday: Turn in: Turn in problems 30, 31, 33, 34, 35, 38, 39, 40, and 41 by Monday evening. As always there will be a cusion, but you really should be done by Monday. Work: Let $\mathfrak c$ be the subspace of $\ell^\infty$ of sequences $x=(x_n)$ such that $\lim_n x_n$ exists and let $\mathfrak c_0$ be the subspace of $\mathfrak c$ for which the limit is $0$. If $y\in \ell^1$, then let $\phi_y$ be the functional on $\mathfrak c_0$ given by \[ \phi_y(x)=\sum_{n=1}^\infty x_ny_n.\] Show that $y\mapsto \phi_y$ is an isometric isomorphism of $\ell^1$ onto $\mathfrak c_0^*$. Describe the dual of $\mathfrak c$. Is either $\mathfrak c_0$ or $\mathfrak c$ reflexive?

Friday: Work: Show that $X$ is reflexive if and only if $X^*$ is. (This is amusing. We always have a chain of isometric injections $X\mapsto X^{**}\mapsto X^{****} \mapsto X^{******} \cdots $. This result shows that either the first arrow (and all subsequent arrows) is a surjection, or none of the arrows is surjective.) Let $\beta\subset \mathcal P(X)$ be a cover of $X$. Show that $\beta$ is a basis for $\tau(\beta)$ if and only if given $U$ and $V$ in $\beta$ and $x\in U\cap V$ there is a $W\in\beta$ such that $x\in W\subset U\cap V$. If $X$ is a finite dimensional normed space, show that the weak topology is the same as the norm topology. (I suggest using the dual basis.) Show that if $X$ is an infinite dimensional normed space, then every nonempty weakly open set is unbounded. (In addition to showing that the topologies are different, this also implies that $x\mapsto \|x\|$ is not weakly continuous.) I suggest showing that given $\phi_1,\dots,\phi_n\in X^*$, then $\bigcap \ker \phi_i \not=\emptyset$. OP: A topological space $(X,\tau)$ is called Hausdorff if given $x\not=y$ in $X$ there are open neighborhoods $U$ and $V$ of $x$ and $y$, respectively, such that $U\cap V=\emptyset$. Prove that the weak topology on a normed space $X$ is Hausdorff.

Week of May 5 to May 9
Assignments Made on:

Monday: Work: Let $f:(X,\tau)\to (Y,\sigma)$ be a function between topological spaces. Show that $f$ is continuous if and only if $f$ takes convergent nets to convergent nets. That is, $f$ is continuous if and only if given $x_\lambda\to x$ in $X$ we have $f(x_\lambda)\to f(x)$ in $Y$. Let $X$ be a normed vector space. Show that a net $(x_\lambda)$ converges to $x$ weakly if and only if $\phi(x_\lambda)\to \phi(x)$ for all $\phi\in X^*$. Does a weakly convergent net $(x_\lambda)$ have to be bounded? OP: Let $S$ be a subset of a vector space $V$. Define $\operatorname{conv}(S)$ to be the collection of sums of the form $\sum_{k=1}^n \lambda_k x_k$ such that $n\ge1$, $x_k\in S$, $\lambda_k\ge0$ and $\sum_{k=1}^n \lambda_k=1$. Show that $\operatorname{conv}(S)$ is the smallest convex subset of $V$ containing $S$. We call $\operatorname{conv}(S)$ the convex hull of $S$.

Wednesday: Work: No lecture today or tomorrow this week.

Friday: Work: Let $(x_\lambda)$ be a net in a compact space $X$. Show that $(x_\lambda)$ has an accumulation point. I suggest letting $F_{\lambda_0} =\overline{\{\, x_\lambda:\lambda\ge\lambda_0\,\}}$ and looking at $x\in \bigcap_\lambda F_\lambda$. (You should compare this to the corresponding proof in metric spaces. And yes, the converse holds. If every net in $X$ has an accumulation point, then $X$ is compact.) Set $(x_n)$ be a sequence in a metric space $X$. Show that $x$ is an accumulation point of $(x_n)$ if and only if $(x_n)$ has a subsequence converging to $x$.

Week of May 12 to 16
Assignments Made on:

Monday: Turn in: Please turn in 42, 43, 44, 45, 46, 48, 49, 51, and 52 by Monday afternoon, May 19th. Announcements: Since folks are often not in lecture at the beginning, here are some things to keep in mind. Homework 4 is due Monday (the 19th). As usual there is a cushion until Tuesday, but you really should have plenty of time to finish working these. We will not meet in our x-hour this week. However, we will likely use our x-hour next week. I have added a brief write up of the product topology and Tychonoff's Theorem and placed it in the Supplements folder in our Files tab on our Canvas page. It also includes a proof of the Jordan -- von Neumann Theorem we will mention later in the week. The solutions file has been updated through Homework 3. I hope every one is using these -- even if you get full credit on the problems. The problems, including the optional ones, can fill important gaps in the lectures. In some cases this material is necessary to a full understanding and in others I would hope it would still allow for importnat insights. Work: No new assignment today.

Wednesday: Work: OP: (A trip back to undergraduate linear algebra.) Let $W_{1}$ and $W_{2}$ be subspaces of a vector space $V$. Recall that the smallest subspace of $V$ containing $W_{1}\cup W_{2}$ is \[ W_{1}+ W_{2} := \{\,w_{1}+w_{2}:w_{j}\in W_{j}\,\}. \] We say that $V$ is the direct sum of $W_{1}$ and $W_{2}$ -- written $V=W_{1}\oplus W_{2}$ -- if $V=W_{1}+W_{2}$ and $W_{1}\cap W_{2}=\{0\}$. Show that $V$ is the direct sum of $W_{1}$ and $W_{2}$ if and only if every $v\in V$ can be uniquely expressed as $v=w_{1}+w_{2}$ with $w_{j}\in W_{j}$. Show that if $V=W_{1}\oplus W_{2}$ then the function $P:V\to V$ given by $P(v)=w_{1}$ if $v=w_{1}+w_{2}$ with $w_{j}\in W_{j}$ is a linear map such that $P^{2}=P$. ($P$ is called the projection of $V$ onto $W_{1}$ along $W_{2}$.) Let $V=(\mathbf R^{2},\|\cdot\|_{2})$, $W_{1}$ the span of $(1,0)$ and $W_{2}$ the span of $(n,1)$ with $n\in \mathbf N$. Then $\mathbf R^{2}=W_{1} \oplus W_{2}$. Let $P$ be the projection of $\mathbf R^{2}$ onto $W_{1}$ along $W_{2}$. Show that $\|P\|\ge n$. (In fact, $\|P\|=\sqrt{n^{2}+1}$, but here we just want to see that we do not always have $\|P\|=1$.) Suppose that $X$ is a reflexive Banach space. Show that the unit ball $B=\{\,x\in X:\|x\|\le 1\,\}$ is weakly compact. (Hint: I suggest showing that the natural map $\iota:X\to X^{**}$ is a homeomorphism from $X$ with the weak topology to $X^{**}$ with the weak-$*$ topology.)

Friday: Work: OP: Let $X$ be a finite-dimensional Banach space. Show that every nonempty closed subset of $X$ has a element of smallest norm. (Hint: In $\mathbf F^n$, bounded sequences have convergent subsequences.) OP: Let $M$ be the set of nonnegative functions $f$ in $\bigl(C([0,2]),\|\cdot\|_\infty \bigr)$ such that \[\int_0^1 f-\int_1^2f=1.\] Show that $M$ is a closed, convex subset that has no element of smallest norm. OP: Show that in a reflexive Banach space, every non-empty closed, convex set $C$ has an element of smallest norm. I suggest using problem #54 to conclude that if $d=d(0,C)$, then $B_n=\{\,x\in X:\|x\|\le d+\frac1n\,\}$ is weakly compact. Let $A_n=B_n\cap C$ and note that $\{A_n\}$ has the FIP. OP: Let $e_n$ be the standard basis element in $\ell^2$. Show that $C=\{\,(1+\frac1n)e_n:n\in\mathbf N\,\}$ is a closed subset of $\ell^2$ with no element of smallest norm. Let $E$ be a nonempty subset of a Hilbert space $H$. Let $Y$ be the subspace spanned by $E$. Then $E^{\perp\perp}$ is the closure of $Y$ in $H$.

Week of May 19 to 23
Assignments Made on:

Monday: Work: This week we will meeting only Monday, Wednesday, and Friday. Life -- aka a contractor -- has made meeting in our x-hour difficult. Let $X=\ell^2$. Show that the sequence $(e_n)$ of standard basis vectors converges weakly to $0$. (Here is another example to see that the norm is not well behaved with respect to the weak topology.)

Wednesday: Work: Let $H$ be a hilbert space. If $x,y\in H$, define $\Theta_{x,y}:H\to H$ by $\Theta_{x,y}(z)=(z\mid y)x$. Compute the norm of $\Theta_{x,y}$ and its adjoint $\Theta_{x,y}^*$. OP: Let $H$ be a separable Hilbert space. Then $T\in \mathcal L(X)$ is called (orthogonally) diagonalizable if there is an orthonormal basis of eigenvectors $\{\,e_n\,\}$ for $T$. Show that $T^*$ is orthognally diagonalizable and that $T$ is normal. OP: Some technical niceities. Let $V$ be a complex vector space. Let $V^o$ be the same additive group and $\iota:V\to V^o$ the identity map. Define scalar multiplication on $V^o$ by $\lambda\cdot \iota(v)= \iota(\overline{\lambda}v)$. Then $V^o$ is a complex vector space called the conjugate space to $V$. If $H$ is a Hilbert space, show that $H^o$ is a Hilbert space which is isometrically isomorphic to $H^*$. Let $P\in\mathcal L(H)$ be the orthogonal projection onto a nonzero subspace $W$. Show that $P=P^*=P^2$ and that $\|P\|=1$. Conversely, show that if $P\in\mathcal L(H)$ and $P=P^*=P^2$, then $P$ is the orthogonal projection onto its range.

Friday: Work: (Dini's Theorem) Suppose that $X$ is a compact metric space and that $C(X)$ is the Banach space of real-valued functions on $X$. Show that if $(f_n)\subset C(X)$ is such that there is a $f\in C(X)$ such that $f_n(x)\nearrow f(x)$ for all $x\in X$, then $f_n\to f$ in $C(X)$. Equivalently, show that $f_n\to f$ uniformly on $X$. (There are probably lots of ways to do this problem, but for any $\epsilon>0$, I suggest letting $E_{n}=\{\, x\in X: |f(x)-f_{n}(x)|<\epsilon$\}.) A linear map $V:H\to H$ is called an isometry if $\|V(x)\|=\|x\|$ for all $x\in H$. Show that the following are equivalent. $V$ is an isometry. $\bigl( V(x) \mid V(y)\bigr)=(x\mid y)$ for all $x,y\in H$. $V^*V=I$. A surjective isometry $U:H\to H$ is called a unitary. Show that the following are equivalent for $U\in \mathcal H$. $U$ is a unitary. $U$ is invertible with $U^{-1}=U^*$. If $\{e_n\}$ an orthonormal basis for $H$, then $\{U(e_n)\}$ is an orthonormal basis for $H$. (Remark, (c) imples (a) is not true unless we know $U$ is both linear and bounded.)

Week of May 28 to May 30
Assignments Made on:

Monday: Work: No Class in honor of Memorial Day

Wednesday: Turn in: Please turn in problems 54, 59, 60, 61, 64, 65, 66, and 67 preferably by Monday but at least by the end of term on Wednesday. This is the last formal homework to be collected. There will be some sort of final assignment available on Wednesday. I will continue to make assignments. Most of them will not be collected, but some could end up on the final assignment. Work: Be aware that we are meeting in our x-hour tomorrow. OP: Let $H=\ell^2(\mathbf Z)$ so that $\{\,e_n:n\in\mathbf Z\,\}$ is an orthonormal basis. Show that $U(e_n)=e_{n+1}$ defines a unitary $U\in\mathcal L(H)$ and that $U$ has no eigenvectors. OP: More fun from the past) Let $H$ be a finite-dimensional complex Hilbers space. Suppose that $T\in\mathcal L(H)$ is normal. Show that $H$ has an orthonormal basis of eigenvectors for $T$. (Since we're working over $\mathbf C$, we know that $T$ has at least one eigenvector $v$. Let $W=\mathbf C \cdot v$. Agrue that $W^\perp$ is invariant for both $T$ and $T^*$ (using problem 70 which should have come before this problem), and that the restriction $T|_{W^\perp}$ of $T$ to $W^\perp$ is a normal operator on $W^\perp$. Now use induction.) Let $T$ be a normal operator. Show that $v$ is an eigenvector for $T$ with eigenvalue $\lambda$ if and only if $v$ is an eigenvector for $T^*$ with eigenvalue $\overline \lambda$. (Everything you every wanted to know about partial isometries.) We call $U\in\mathcal L(H)$ a partial isometry if there is a closed subspace $E$ on which $U$ is isometric and $U(E^\perp)=\{0\}$. Suppose $U$ is a partial isometry (on $E$ as above) and $P:=U^*U$. Observe that $$(P(x)\mid x)=\|x\|^2.$$ Conclude that $P(x)=x$ and that $P$ is the orthogonal projection onto $E$. (First use Cauchy-Schwarz to show $\|P(x)\|=\|x\|$.) If $U$ is a partial isometry, then show that $U=UU^*U$. (Hint: $U-UU^*U=U(I-P)$.) Conversely, show that if $V\in\mathcal L(H)$ is such that $V^*V$ is a projection, then $V$ is a partial isometry. Show that if $U$ is a partial isometry, then $U^*$ is a parital isometry with space the range of $U$. Describe the sense in which $U$ and $U^*$ are inverses to each other. (Think of $U$ as a operator from its space $E$ onto its range.)

Thursday (x-hour): Work: No New Assignment

Friday: Work: It $T\in\mathcal L(H)$, then the spectrum $\sigma(T)$ is the set of $\lambda\in\mathbf C$ such that $T-\lambda I$ is not invertible. If $\dim(H)<\infty$, then $\lambda\in\sigma(T)$ if and only if $\lambda$ is an eigenvalue of $T$. This is not longer true if $\dim(H)=\infty$ (see optional problem #75). Here I want you to prove that if $T$ is normal and $\lambda\in\sigma(T)$ then $\lambda$ is an approximate eigenvalue in the sense that there is a sequence $(x_n)$ of unit vectors in $H$ such that $(T-\lambda I)(x_n)$ tends to $0$ in norm. (Hint: Recall that we proved a normal operator in invertible if and only if it is bounded away from $0$.) Show that $T$ is compact if and only if $|T|$ is compact. (Recall that if $U|T|$ is the polar deomposition of $T$ then $U^*U$ is the orthogonal projection onto the space of $U$ which is the closure of the range of $|T|$. Use this to show $U^*T=|T|$.) Show that if $I$ is any (not necessarily closed) nonzero ideal in $\mathcal L(H)$ then $\mathcal{L}_f(H)\subset I$. In other words, the finite rank operators are a mininmal ideal in $\mathcal L(H)$. (Show $\mathcal {L}_f$ has nontivial intersection with $I$ and that $\mathcal{L}_f(H)$ has no nontrivial proper ideals.) OP: (This problem involves measure theory, so I've included it for interest only. It will not appear on the final.) Let $H=L^2([0,1])$ and define $T:H\to H$ by $T(h)(x)=xh(x)$. Show that $\|T\|=1$, $T\ge 0$, and that $T$ has no eidenvectors even though $\sigma(T)=[0,1]$. Hence $T$ is not diagonalizable.

Week of June 2 to 4
Assignments Made on:

Monday: Work: In lecture, we saw that a linear map $T:H\to H$ whose restriction to the unit ball is weak-norm continuous was a compact operator. Here we want to see that if $T:H\to H$ is weak-norm continuous, then $T$ is a finite-rank operator. (Observe that for such a $T$, $x\mapsto \|T(x)\|$ is weakly continuous. Hence there are $x_1,\dots,x_n\in H$ such that $|(x\mid x_k)|<1$ for all $k$ implies that $\|T(x)\|<1$.)

Wednesday: Work: Final Lecture

Friday: Work:

Dana P. Williams
Last updated June 01, 2025 16:53:10 EDT