Math 113
Functional Analysis
Last updated May 26, 2017 09:01:15 EDT

Announcements:

• Our first class meeting is Monday, March 27, 2017, at 2:10pm.

### Homework Assigments

 Monday: Work: We want to show that the metrics induced on $\mathbf {R}^n$ by the $p$-norms are all strongly equivalent for $1\le p \le\infty$. Observe that it will suffice to show that given $1\le p\le\infty$ there are $c,d>0$ such that $$c\|\mathbf x\|_2\le \|\mathbf x\|_p \le d\|\mathbf x\|_2\tag{*}$$ for all $\mathbf x\in\mathbf{R}^n$. Prove (*) using the fact that a continuous function on a closed bounded subset $C$ of $\mathbf{R}^n$ attains its maximum and minimum on $C$. (Why is "attains" important above?) Give an example of a metric on $\mathbf{R}^n$ which is not equivalent to any of the metics in problem 1. Wednesday: Work: Let $E$ be a subset of a metric space $X$. We say that $x$ is a limit point of $E$ if there is a sequence $(x_n)\subset E$ such that $x=\lim_{n\to\infty} x_n$. Show that $E$ is closed if and only if $E$ contains all its limit points. State and prove a result characterizing open sets in a metric space interms of sequences (as we did for closed sets in the previous problem). The following terminology might be useful. If $U$ is a subset of a metric space $X$, then a sequence $(x_n)\subset X$ is eventually in $E$ if there is a $N$ such that $n\ge N$ implies $x_n\in E$. Let $\rho$ and $\sigma$ be metrics on $X$. Show that $\rho$ and $\sigma$ are equivalent if and only if they have the same convergent sequences. That is, show that $x_n\to x$ with respect to $\rho$ if and only if $x_n\to x$ with respect to $\sigma$. Consider $L^1(X)$ for a measure space $(X,\mathfrak M,\mu)$. Let $U$ be the subset of $f\in L^1(X)$ such that $$\int_X \mathop{\rm Re}(f(x))\,d\mu(x)<1.$$ Show that $U$ is open (with respect to the metric induced by $\|\cdot\|_1$). Friday, April 1: Work: Let $(X,\rho)$ be a metric space. If $A\subset X$, then define $\rho(x,A)=\inf\{\, \rho(x,y):y\in A\,\}$. Show that $\rho(x,A)=0$ if and only if $x\in \overline A$. Show that $x\mapsto \rho(x,A)$ is continuous. Show that if $A$ and $B$ are disjoint nonempty closed subsets of $X$, then there is a $f\in C_b(X)$ such that (i) $0\le f(x)\le 1$ for all $x$, (ii) $f(x)=1$ if and only if $x\in A$, and (iii) $f(x)=0$ if and only if $x\in B$. (Hint: try $\rho(x,B)/(\rho(x,A)+\rho(x,B))$.) Show that a Cauchy sequence in a metric space with a convergent subsequence is necessarily convergent. Let $X$ be a metric space. Prove that the uniform limit of continuous functions $f_n:X\to \mathbf C$ is continuous. Let $X$ be a metric space. Recall that we say $f:X\to \mathbf C$ is bounded if $\|f\|_\infty<\infty$. A sequence $(f_n)$ of functions $f_n:X\to\mathbf D$ is uniformly bounded if there is a $M$ such that $\|f_n\|_\infty\le M$ for all $n$. Also, $(f_n)$ is called uniformly Cauchy if for all $\epsilon>$ there is $N$ such that $n,m\ge N$ implies $|f_n(x)-f_m(x)|<\epsilon$ for all $x\in X$. Show that a uniformly Cauchy sequence $(f_n)$ of bounded functions is uniformly bounded. In particular, $(f_n)$ converges to a bounded function.

Week of April 3 to 7
 Monday April 3: Study: We are working through parts of chapters 9 and 10 of Royden (and Fitzpatrick). Do: 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 $X$. Let $(x_n)$ be a sequence is a complete metric space $(X,\rho)$. Suppose that $\rho(x_n,x_{n+1})<1/2^n$ for each $n$. Conclude that $(x_n)$ is convergent. What if instead we have $\rho(x_n,x_{n+1})<1/n$? Let $E=(0,\infty)$. Let $d$ be the usual metric on $E$. Define $\delta(x,y)=\bigl|\frac1x-\frac1y\bigr|$. Show that $\delta$ is a metric on $E$ equivalent to $d$. Is the map $x\mapsto \frac1x$ uniformly continuous from $(E,d)$ to $(E,d)$? What about from $(E,d)$ to $(E,\delta)$? Is $(E,\delta)$ complete? What about $((0,1],\delta)$? 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.) Wednesday April 5: Study: Do: 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 $\bigcap_{F\in\mathcal F}F\not=\emptyset$. Show that $E\subset X$ is totally bounded if and only if there is an $\epsilon$-net for $E$ for all $\epsilon >0$. 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 $X=(0,1)$. 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.) Friday April 7: Study: Do: 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.) (Missing problem number) Show that a subset of a compact metric space is compact if and only if it is closed. 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 $-f$.)

Week of April 10 to 14
 Monday: Study: The previous week's assignments (11-23) are due today. Do: Let $K$ be a compact subset of a metric space $X$ and let $K \subset U$ be open. Show that there is an open set $V$ such that $K\subset V\subset \overline V\subset U$. (Consider homework problem #7.) 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 obvious complete product metric. It suffices to see that that the range of $\phi$ is closed. Wednesday: Study: Do: 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 $\mathbf R$. 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 #25.) 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. 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, $\|\overline T\|=\|T\|$. 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 $y$.) Friday: Study: Problems 24 to 43 are due WEDNESDAY. I'll be posting selected homework solutions (Last modified December 31, 1969) from time to time. Keep an eye on the date stamp for new additions. In particular, I may update the solutions for the previous week as I see where folks had trouble when I get around to grading them. Do: 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 17 to 21
 Monday: Study: Do: 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 $$\inf\{\, \|T(x)\|: \|x\|=1\,\}>0.$$ Let $X$ be compact metric space and $A$ a closed subspace of $C(X)$ and let $E$ be closed in $X$. Suppose each $g \in C(E)$ has an extension to a $f\in A$. (That is, $f|_E=g$.) Show that there is a constant $M>0$ such that for every $g\in C(E)$ we can find an extension $f$ such that $\|f\|_\infty\le M\|g\|_\infty$. Wednesday: Study: Do: 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 $\psi(x)=\phi(x)-i\phi(ix)$. Let $\{e_\lambda\}_{\lambda\in\Lambda}$ be a Hamel basis for a Banach space $X$. This means that every $x\in X$ can be written uniquely as $x=\sum_\lambda c_\lambda e_\lambda$ with only finitely many $c_\lambda$ nonzero. Hence we get dual functionals $e_\lambda^*$ given by $e_\lambda^*(x)=c_\lambda$. Show that at most finitely many of these dual functionals can be continuous. Let $S=\{\,e_\lambda^*:\text{$e_\lambda^*$is continuous}\,\}$. For each $e_\lambda^*\in S$, choose a constant $a_\lambda >0$. Use the PUB to show that $S=\{\,a_\lambda \cdot e_\lambda^*:\text{$e_\lambda^*$is continuous}\,\}$ is bounded in $X^*$. (Of course, the bound will depend on the choice of the $a_\lambda$.) Conclude that $S$ must be finite. Friday: Study: Here are selected homework solutions (Last modified December 31, 1969) up through assignment 3. Do: 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 $\Phi(\iota_1(x))=\iota_2(x)$. (This allows to abuse language slightly and talk about the completion of $X$.) Recall that we write $\mathcal c$ for the subspace of $\ell^\infty$ of bounded sequences $(x_n)$ such that $\lim_n x_n$ exists and $\mathcal c_0$ for the subspace of $\mathcal c$ for the which the limit is zero. If $y=(y_n)\in\ell^1$, then we get a linear functional $\phi_y$ on $\mathcal c_0$ given by $$\phi_y(x)=\sum_n x_ny_n.\tag{*}$$ Show that $y\mapsto \phi_y$ is an isometric isomorphism of $\ell^1$ onto ${\mathcal c_0}^*$. (In this problem, I found it convenient to introduce the function $\operatorname{sgn}:\mathbf C\to \mathbf C$ given by $\operatorname{sgn}(z)=\frac z{|z|}$ if $z\not=0$ and $0$ otherwise.) If instead, we let $y\in\ell^\infty$, then the formula in $(*)$ gives us a linear functional on $\ell^1$. Show that in this case $y\mapsto \phi_y$ is an isometric isomorphism of $\ell^\infty$ onto ${\ell^1}^*$. Describe the dual of $\mathcal c$. Are either $\mathcal c_0$ or $\mathcal c$ reflexive? 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 $\liminf_n x_n \le L(x) \le \limsup_n x_n$. 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 $Y$. 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.)

Week of April 24 to 28
 Monday: Study: Here are selected homework solutions (Last modified December 31, 1969). Note the time stamp. Please let me know if these have not been updated to what has currently been turned in. As agreed, the assignment ending with problem 153 is due Wednesday. Let's just start out assuming this week's assignment will be due a week from Wednesday and get it over with. Do: 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 $x\in W\subset U\cap V$. Wednesday: Study: Do: 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 $\bigcap \ker \phi_i \not=\emptyset$. 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. Not to be turned in: The product toplogy is one of the more ubiquitous objects in elementary topology. Let $(X_a,\tau_a)$ be a topological space for all $a\in A$. Recall that the Cartesian product $\prod_{a\in A}X_a$ is the set of all functions $x:A\to \bigcup_{a\in A}X_a$ such that $x(a)\in X_a$. If $a_0\in A$, then the projection $p_{a_0}$ onto the $a_0$-factor is the map $p_{a_0}:\prod_{a\in A} X_a\to X_{a_0}$ given by $p_{a_0}(x)=x(a_0)$. The product topology on $\prod_{a\in A} X_a$ is the initial topology induced by the projections maps. Thus the product topology is the smallest topology on the product such that each projection is continuous. A subbasis is given by the sets $U(a,V)=p_a^{-1}(V)$ for any $a\in A$ with $V$ open in $X_a$. Let $(x_\lambda)$ be a net in $\prod_{a\in A}X_a$. Then $x_\lambda \to x$ in the product topology if and only if $x_\lambda(a)\to x(a)$ for all $a\in A$. (So the product topology can be thought of as the topology of pointwise convergence.) The Tychonoff Theorem asserts that the (arbitrary) product of compact spaces is compact in the product topology. We'll use this to prove the Alaoglu Theorem in due course. Right now, I want to point out that #229 does not hold in general topological spaces. For each $\alpha\in\ell^\infty$, let $D_\alpha$ be a closed disk in $\mathbf C$ such that $\alpha_n\in D_\alpha$ for all $n\ge 1$. Then $Z=\prod_{\alpha\in \ell^\infty} D_\alpha$ is compact in the product topology. Let $(z_n)\subset Z$ be the sequence given by $z_n(\alpha)=\alpha_n$. Then $(z_n)$ has accumulation points (just because $Z$ is compact and applying #228), but no converent subsequences. Friday: Study: Do: 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 $S$. 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? 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 $x$. Let $X=\ell^2$. Show that the sequence $(e_n)$ converges weakly to $0$. (As usual, $e_n=\delta_n$. Here is another example to see that the norm is not well behaved with respect to the weak topology.)
 Monday: Study: Do: Wednesday: Study: Here are selected homework solutions (Last modified December 31, 1969) up through assignment 5. Do: Let $\bigl(V,(\cdot\mid\cdot)\bigr)$ be pre-inner product space. Let $N=\{\,v\in V:\|v\|=0\,\}$ be the subspace of length zero vectors. Show that $$(v+N\mid w+N)=(v\mid w)$$ is a well-defined inner product on $V/N$. Let $\bigl(V,(\cdot\mid \cdot)\bigr)$ be an inner product space. We want to see that its completion $\widetilde V$ is a Hilbert space without invoking the Jordan-von Neumann Theorem. I suggest the following. Let $i:V\to \widetilde V$ be the natural map. Show that if $(v_{n})$ and $(w_{n})$ are Cauchy in $V$, then $\bigl((v_{n}\mid w_{n})\big)$ is Cauchy in $\mathbf F$. Show that if $i(v_{n})$ and $i(v_{n}')$ converge to $v$ in $\widetilde V$ while $i(w_{n})$ and $i(w_{n}')$ converge to $w$ in $\widetilde V$, then $\lim_{n} (v_{n}\mid w_{n}) = \lim_{n}(v_{n}'\mid w_{n}')$. (Both the limits exist by the previous part.) Show there is a function $(\cdot\mid\cdot)_{0}$ on $\widetilde V\times\widetilde V$ such that $(v\mid w)_{0}=\lim(v_{n}\mid w_{n})$ whenever $i(v_{n})\to v$ and $i(w_{n})\to w$ in $\widetilde V$. Show that $(\cdot\mid\cdot)_{0}$ is an inner product on $\widetilde V$ such that $i:V\to\widetilde V$ is inner product preserving. In particular, $\|v\|=(v\mid v)_{0}^{\frac12}$ for all $v\in \widetilde V$. 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 $H$. 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 $H^*$ Friday: Study: Do: Let $E$ and $F$ be closed subspaces of a Hilbert space $H$ with $\dim E<\infty$ and $\dim E<\dim F$. Show that $E^\perp\cap F\not=\{0\}$. Let $\{v_n\}$ be a countable linearly independent set in a Hilbert space $H$. Find a countable orthonormal set $\{e_n\}$ such that for each $n\ge1$ we have $\operatorname{span}\{v_1,\dots,v_n\}= \operatorname{span}\{e_1,\dots,e_n\}$. (Proceed inductively and consider the projection of $v_{n+1}$ onto $\operatorname{span}\{e_1,\dots,e_n\}$. Or just look up Gram-Schmidt in your linear algebra texts.) Let $\{e_n\}$ be an orthonormal basis for a Hilbert space $H$. Show that there is a $T\in\mathcal{L}(H)$ such that $T(e_n) = (1+n^{-1})e_n$. (In other words, $T(x)=\sum_n (1+n^{-1})(x\mid e_n)e_n$.) Check that $$\{\,x\in H:\|T(x)\|\le 1\,\}:=C= \{\, x\in H: \sum_n (1+n^{-1})^2|(x\mid e_n)|^2\le 1\,\}.$$ Conclude that $C$ is a nonempty, closed convex subset. Observe in contrast to our Key Lemma on convex subsets of Hilbert space, $C$ has no element of largest norm. (Hint: note that $\|x\|=\sum_n|(x\mid e_n)|^2< \sum_n(1+n^{-1})^2|(x\mid e_n)|^2$.) Let $H=\mathbf R^2$. Note that $H$ is the direct sum of the dimensional subspaces $V$ and $W$ spanned by $(1,0)$ and $(n,1)$ respectively (for a fixed $n\ge1$). Let $P=P_{V,W}$ be the projection of $H$ onto $V$ along $W$. Show that $\|P\|=\sqrt{n^2+1}$. Suppose that $H_n$ is a Hilbert space for each $n\ge1$. As in lecture, let $\bigoplus_{n=1}^\infty H_n$ be the Hilbert space completion of the algebraic direct sum $\sum_{n=1}^\infty H_n$ (see problem \#301). The purpose of this problem is to verify that $\bigoplus_{n=1}^\infty H_n$ can be identified with $$H= \Bigl\{\, h\in\prod H_n:\sum_{k=1}^\infty \|h(k)\|^2<\infty\,\Bigr\}.$$ I suggest the following. Observe that it suffices to see that $H$ is a Hilbert space, and that for that, it suffices to see that $H$ is a complete. Observe that $\|h_n - h_m\|_2^2\le \epsilon^2$ if and only if $\sum_{k=1}^N \|h_n(k) -h_m(k)\|^2\le\epsilon^2$ for all $N\ge 1. Conclude that$\|h_n-h_m\|_2^2\le \epsilon^2$for all$n\ge M$implies$\sum_{k=1}^N\|h(k)-h_m(k)\|^2\le \epsilon^2$for all$N\ge1$. Conclude that$h$belongs to$H$and that$h_n\to h$in$H$. Week of May 8 to 12 Assignments Made on:  Monday: Study: Do: For ease of exposition, let's assume$H$is a separable complex Hilberts space in these problems. A linear map$V:H\to H$is called an isometry if$\|V(x)\|=\|x\|$for all$x\in H$. Show that$V$is an isometry if and only if$\bigl( V(x) \mid V(y)\bigr)=(x\mid y)$for all$x,y\in H$. A linear map$U:H\to H$is called a unitary (or a Hilbert space isomorphism) if$U$is a bijection such that$\bigl( U(x) \mid U(y) \bigr)=(x\mid y)$for all$x,y\in H$. Show that the following are equivalent.$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$. (A step back in time.) Let$H$be a finite-dimensional Hilbert space with orthonormal basis$\sigma=\{e_1,\dots,e_n\}$. If$T\in \mathcal{L}(H)$, let$[T]$be the matrix of$T$with respect to$\sigma$. Show that if$(a_{ij})=[T]$, then$a_{ij}=(T(e_j)\mid e_i)$. Show that$[T^*]$is the conjugate transpose of$[T]$. Show that$T$is unitrary if and only if the columns of$[T]$for an orthonormal basis for$H$. Wednesday: Study: Here are selected homework solutions (Last modified December 31, 1969) up through assignment 6. Do: (Dini's Theorem) Suppose that$X$is a compact metric space and$(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$. Show that$f_n\to f$in$C(X)$. Equivalently, show that$f_n\to f$uniformly on$X$. (There are probably lots of ways to do this problem. But I found it helpful to note that since the convergence is monotonic, if$f_n\not\to f$uniformly, then there is a$\epsilon_0>0$such that$\|f-f_n\|_\infty\ge\epsilon_0$for all$n$.) Suppose that$P$is the orthogonal projection onto the closed subspace$E$of$H$. We already know that$P$is linear with$\|P\|=1$(provided$E\not=H$). Show that$P$is a positive operator. Conversely, if$P$is any self-ajoint idempotent in$\mathcal L(H)$(i.e.,$P^2=P$), then show that$P$is the orthogonal projection onto its range. Let$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$. Let$E$be a closed subspace of a (separable) Hilbert space$H$. Let$A$be an orthonormal basis for$E$and$B$an orthonormal basis for$E^\perp$. Show that$C=A\cup B$is an orthonormal basis for$H$. What does this say about the$\dim E$and$\dim E^\perp$when$H$is finite dimensional? (More fun from the past) Let$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^*$and that the restriction$T|_{W^\perp}$of$T$to$W^\perp$is a normal operator on$W^\perp$. Now use induction.) (Everything you every wanted to know about partial isometries.) We call$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.) Friday: Study: Do: 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.) 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$\|T(x)\|<1$.) Show that a norm-weak continuous map is actually norm-norm continuous. Week of May 15 to 19 Assignments Made on:  Monday: Study: Catch up Do: Wednesday: Study: Do: Let's review the final details of the Spectral Theorem for Normal Compact Operators. Let$T$be a normal compact operator on$H$. Let$\alpha=\{e_k\}$be a set of orthonormal eigenvectors for$T$, and let$P$be the orthogonal projection onto$E=\overline{\operatorname{span}\{e_k\}}$. Show that$P$and$T$commute. Let$S=(I-P)T$. Show that$S$is compact and normal. If$S=0$, then show that any unit vector$e_0\in E^\perp$is an eigenvector for$T$. If$S\not=0$, then our partially proved lemma implies that$S$has en eigenvector$e_0$such that$S(e_0)=\lambda e_0$with$|\lambda|=\|S\|$. Show that$e_0 \in E^\perp$and that$e_0$is an eigenvector for$T$. Conclude that$\alpha\cup\{e_0\}$is an orthonormal set of eigenvectors for$T$. Show that a diagaonalizable operator$T\in\mathcal{L}(H)$is normal. How is the norm of$T$related to its eigenvalues? Let$I$be a closed ideal in a Banach algebra$A$. Show that the quotient norm satisfies$\|q(x)q(y)\|\le \|q(x)\|\|q(y)\|$. (Thus$A/I$is a Banach algebra.) Let$\Omega$be a connected open subset of$\mathbf C$. (We call$\Omega$a domain.) If$A$is a Banach algebra, then we call$f:\Omega\to A$strongly holomorphic if$f'(z):=\lim_{h\to 0}h^{-1}(f(z+h)-f(z))$exists (in$A$) for all$z\in \Omega$. Observe that if$f$is strongly holomorphic and$\phi\in A^*$, then$\phi\circ f$is holomorphic on$\Omega$in the usual sense. Show that if$f$is strongly holomorphic on$\mathbf C$and bounded, then$f$is constant. Friday: Study: Here are selected homework solutions (Last modified December 31, 1969) up through assignment 7. Do: Let$X$be a compact metric space and let$J$be a closed ideal in$C(X)$. Suppose there is a$f\in J$such that$f(x)\not=0$for all$x\in X$. Show that$J=C(X)$. Suppose that for each$x\in X$there is a$f\in J$such that$f(x)\not= 0$. Show that$J=C(X)$. Conclude that every maximal ideal in$C(X)$is of the form$J_x=\{\, f\in C(X):f(x)=0\,\}$for some$x\in X$. Conclude that every$h\in \Delta$is of the form$h(f)=f(x)$for some$x\in X$. Let$h_x$be evaluation at$x$as above. Show that$x\mapsto h_x$is a homeomorphism of$X$onto$\Delta$. (Hint: a continuous bijection$f:X\to Y$between compact metric spaces is automatically a homeomorphism -- show that$f$maps closed sets to closed sets.) Let$A$be the subset of$2\times 2$complex matrices given by $$\Bigl\{ \begin{pmatrix}a & b \\ 0 & a \end{pmatrix}:a,b\in\mathbf{C}\,\Bigr\}.$$ Observe that$A$is a two dimensional unital commutative Banach algebra and that$\operatorname{Rad}(A)\not=\{0\}$. (Hint: any complex homomorphism on$A$is in particular a linear map from$A$to$\mathbf C\$.) FINAL EXAM: Study: Here is the final exam (Last modified December 31, 1969). I will post the final exam here by the end of the week. It is due in my office, email, or my mailbox my noon on Thursday, June 1st. Since there are bound to be typos or clarifications required, I'll post updates here as necessary. Do: