Math 641600 — Fall 2016
Assignments
Assignment 1  Due Tuesday, September 6, 2016.
 Read sections 1.11.4
 Do the following problems.
 Section 1.1: 3(c), 5, 7(a), 8
 Section 1.2: 1(a,b), 8, 9, 10(a,b)
 Let $U$ be a subspace of an inner product space $V$, with the
inner product and norm being $\langle\cdot,\cdot \rangle$ and
$\\cdot\$. Also, let $v$ be in $V$. (Do not assume that
$U$ is finite dimensional or use arguments requiring a basis.)
 Fix $v\in V$. Show that if there exists $p\in U$ such that $p$
satisfies either (i) $\min_{u\in U}\vu\ = \vp\$ or (ii) $vp\in
U^\perp$, then it satisfies both (i) and (ii). Moreover, if $p$
exists, it is unique.
 Suppose $p$ exists for every $v\in V$. Since $p$ is uniquely
determined by $v$, we may define a map $P: V \to U$ via
$Pv:=p$. Show that $P$ is a
linear map and that $P$ satisfies $P^2 = P$. ($P$ is called
an orthogonal projection. The vector $p$ is the orthogonal
projection of $v$ onto $U$.)
 If the projection $P$ exists, show that for all $w,z\in V$,
$\langle Pw,z\rangle = \langle Pw,Pz\rangle= \langle
w,Pz\rangle$. Use this to show that $U^\perp= \{w\in
V\colon Pw=0\}$.
 Suppose that the projection $P$ exists. Show that $V=U\oplus
U^\perp$, where $\oplus$ indicates the direct sum of the two spaces.
 Let $U$ and $V$ be as in the previous exercise. Suppose that $U$
is finite dimensional and that $B=\{u_1,u_2,\ldots,u_n\}$ is an
ordered basis for $U$. In addition, let $G$ be the $n\times n$
matrix with entries $G_{jk}= \langle u_k,u_j\rangle$.
 Let $v\in V$ and $d_k := \langle v,u_k\rangle$. Show that $p$
exists for every $v$ and is given by $p=\sum_j x_j u_j\in U$, where
the $x_j$'s satisfy the normal equations, $d_k = \sum_j
G_{kj}x_j$. (Hint: use exercise 3 above.)
 Show that if B is orthonormal, then $Pv=\sum_j \langle
v,u_j\rangle u_j$.
Assignment 2  Due Tuesday, September 13, 2015.
 Read the notes on
Adjoints & Selfadjoint Operators, and sections 2.1 and
2.2 in Keener.
 Do the following problems.
 Section 1.3: 2, 3
 Section 1.4: 4
 Let $A = \begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 1\\ 1 & 1 &
1\end{pmatrix}$. Find the $QR$ decomposition of $A$.
 Let $\{v_1,\ldots,v_m\}$ be a set of linearly independent vectors
in $\mathbb R^n$, with $m < n$, and let $A := [v_1\ \cdots \ v_m]$;
that is, $A$ is an $n\times m$ matrix having the $v_j$'s for columns.
 Use GramSchmidt to show that $A=QR$, where $Q$ is an $n\times m$
matrix having columns that are orthonormal and $R$ is an invertible,
upper triangular $m\times m$ matrix.
 Let $\mathbf y\in \mathbb R^n$. Use the normal equations for a
minimization problem to show that the minimizer of $\ \mathbf y 
A\mathbf x\$ is given by $\mathbf x_{min} = R^{1}Q^\ast \mathbf y$.
 Let $A$ be an $n\times n$ matrix and let $p_A(\lambda) =
\det(A\lambda I)$. Show that if $A'$ is similar to $A$, then
$p_{A'}=p_A$. In addition, show that the trace of $A$,
$\text{Tr}(A)=\sum_{j=1}^n a_{jj}$, satisfies
$\text{Tr}(A')=\text{Tr}(A)$ and that $\det(A)=\det(A')$.
 (This is a generalization of Keener's problem 1.3.5.) Let $A$ be
a selfadjoint matrix with eigenvalues $\lambda_1\ge
\lambda_2,\ldots,\ge \lambda_n$. Show that for $ 2\le k < n$ we have
\[ \max_U \sum_{j=1}^k \langle Au_j,u_j \rangle =\sum_{j=1}^k
\lambda_j, \]
where $U=\{u_1,\ldots,u_k\}$ is any o.n. set. (Hint: One direction is
trivial. To show that $\max_U \sum_{j=1}^k \langle Au_j,u_j \rangle
\le \sum_{j=1}^k \lambda_j$ do the following. Put $A$ in diagonal
form. Without loss of generality, one may assume that $\lambda_k\ge 0
\ge \lambda_{k+1}$. Choose a matrix $B$ with these properties: for
all $x$, $x^TBx\ge 0$, $x^TBx\ge x^TAx$, and $\text{Tr}(B) =
\lambda_1+\lambda_2+\cdots +\lambda_k$. Add vectors to the $u_j$'s to
get an o.n. basis for $\mathbb R^n$. Use the invariance of the trace
proved in the previous problem to complete the proof.)
 Let U be a unitary, n×n matrix. Show that the following hold.
 < Ux, Uy > = < x, y >
 The eigenvalues of U all lie on the unit circle, λ=1.
 Show that U is diagonalizable. (Hint: follow the proof for the
selfadjoint case. The key to this problem is observing that
$Ux=\lambda x$ if and only if $U^*x = \bar \lambda x$.)
Assignment 3  Due Tuesday, September 20, 2016.
 Read sections 2.1, 2.2.1 and 2.2.2 and the notes
on
Banach Spaces and Hilbert Spaces .
 Do the following problems.
 Section 2.1: 3, 5
 Let $k(x,y) = x+ 3x^2y + xy^2$ and $\langle
f,g\rangle=\int_{1}^1 f(x)g(x)(1+x^2)dx$. Consider the operator
$Lu=\int_{1}^1 k(x,y) u(y)dy$. In the notes, we have shown that
$L:P_2\to P_2$.
 Relative to the inner product above, find $L^\ast$ and
$\text{Null}(L^\ast)$.
 Find a condition on $q\in P_2$ for which $Lp=q$ always has a
solution. Is this different from what was in the notes?
 Show that $\ell^2$, under the inner product $\langle
x,y\rangle = \sum_{j=1}^\infty x_j \overline{y_j}$, is a Hilbert
space.
 This problem concerns several important inequalities.
 Show that if α, β are positive and α + β
=1, then for all u,v ≥ 0 we have
u^{α}v^{β} ≤ αu + βv.
 Let x,y ∈ R^{n}, and let p > 1 and define
q by q^{1} = 1  p^{1}. Prove Hölder's
inequality,
∑_{j} x_{j}y_{j} ≤ x_{p}
y_{q}.
Hint: use the inequality in part (a), but with appropriate choices of
the parameters. For example, u =
(x_{j}/x_{p})^{p}
 Let x,y ∈ R^{n}, and let p > 1. Prove
Minkowski's inequality,
x+y_{p} ≤ x_{p} + y_{p}.
Use this to show that x_{p} defines a norm on
R^{n}. (Hint: you will need to use Hölder's
inequality, along with a trick.)
 Let $f\in C^1[0,1]$. Show that
$\f\_{C[0,1]}\le C\f\_{H^1[0,1]}$, where $C$ is a constant
independent of $f$ and $\f\_{H^1[0,1]}^2 := \int_0^1\big( f(x)^2 +
f'(x)^2\big)dx$.
Assignment 4  Due Thursday, September 29, 2016.
 Read the notes on
Lebesgue
integration and
on Orthonormal
sets and expansions.
 Do the following problems.
 Section 2.1: 10, 11
 Section 2.2: 1 (Do $w=1$.), 10
 A measurable function whose range consists of a finite number of
values is a simple function —
see Lebesgue
integration, p. 5. Using the Lebesgue sums in eqn. 2 and the
definition of the Lebesgue integral given in terms of the Lebesgue
sums, show that the integral of a simple function is given by eqn. 3
on p. 6.
 Let F(s) = ∫_{ 0}^{∞} e^{ − s
t} f(t)dt be the Laplace transform of f ∈
L^{1}([0,∞)). Use the Lebesgue dominated convergence
theorem to show that F is continuous from the right at s = 0. That is,
show that
lim_{ s↓0} F(s) = F(0) = ∫_{
0}^{∞}f(t)dt
 Let f_{n}(x) = n^{3/2} x e^{n x}, where
x ∈ [0,1] and n = 1, 2, 3, ....
 Verify that the pointwise limit of f_{n}(x) is f(x) = 0.
 Show that f_{n}_{C[0,1]} → ∞ as n
→ ∞, so that f_{n} does not converge uniformly to
0.
 Find a constant C such that for all n and x fixed
f_{n}(x) ≤ C x^{−1/2}, x ∈ (0,1].
 Use the Lebesgue dominated convergence theorem to show that
lim_{ n→∞} ∫_{ 0}^{1}
f_{n}(x)dx = 0.
 Let $U:=\{u_j\}_{j=1}^\infty$ be an orthonormal set in a Hilbert
space $\mathcal H$. Show that the two statements are
equivalent. (You may use what we have proved for o.n. sets in
general; for example, Bessel's inequality, minimization properties,
etc.)
 $U$ is maximal in the sense that there is no nonzero vector in
$\mathcal H$ that is orthogonal to $U$. (Equivalently, $U$ is not a
proper subset of any other o.n. set in $\mathcal H$.)
 Every vector in $\mathcal H$ may be uniquely represented as the
series $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.
Assignment 5  Due Thursday, October 8, 2016.
 Read sections 2.2.22.2.4 and the notes on
Approximation
of Continuous Functions.
 Do the following problems.
 Section 2.2: 8, 9. Note: the formula in Problem 8(e)
has an $n!$ missing in the numerator. It should be
\[ T_n(x) = \frac{(1)^n2^n
n!}{(2n)!}(1x^2)^{1/2}\frac{d^n}{dx^n}(1x^2)^{n1/2}
\]
 This problem is aimed at showing that the Chebyshev polynomials
form a complete set in $L^2_w$, which has the weighted inner product
\[ \langle f,g\rangle_w := \int_{1}^1
\frac{f(x)\overline{g(x)}dx}{\sqrt{1  x^2}}. \]
 Show that the continuous functions are dense in $L^2_w$. Hint: if
$f\in L^2_w$, then $ \frac{f(x)}{(1  x^2)^{1/4}}$ is in $L^2[1,1]$.
 Show that if $f\in L^\infty[1,1]$, then $\f\_w \le
\sqrt{\pi}\f\_\infty$.
 Follow the proof given in
the notes on Orthonormal
Sets and Expansions showing that the Legendre polynomials form a
complete set in $L^2[1,1]$ to show that the Chebyshev polynomials
form a complete orthogonal set in $L^2_w$.
 Let $\delta>0$. We define the modulus of continuity for $f\in
C[0,1]$ by $\omega(f,\delta) := \sup_{\,\,st\,\,\le\,
\delta,\,s,t\in [0,1]}f(s)f(t)$.
 Explain why $\omega(f,\delta)$ exists for every $f\in C[0,1]$.
 Fix $\delta>0$. Let $S_\delta = \{ \epsilon >0 \colon f(t)  f(s)
< \epsilon \forall\ s,t \in [0,1], \ s  t \le \delta\}$. In other
words, for given $\delta$, $S_\delta$ is in the set of all
$\epsilon$ such that $f(t)  f(s) < \epsilon$ holds for all $s 
t\le \delta$. Show that $\omega(f, \delta) = \inf S_\delta$
 Show that $\omega(f,\delta)$ is non decreasing as a
function of $\delta$. (Or, more to the point, as $\delta \downarrow 0$,
$\omega(f,\delta)$ gets smaller.)
 Show that $\lim_{\delta \downarrow 0} \omega(f,\delta) = 0$.
 Calculus problem: Let g be C^{2} on an interval
[a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
g_{C[a,b]} ≤ (h^{2}/8)
g′′_{C[a,b]}.
Give an example that shows
that $1/8$ is the best possible constant.
 Use the previous problem to show that if f ∈
C^{2}[0,1], then the equally spaced linear spline interpolant
f_{n} satisfies
f −
f_{n}_{C[0,1]} ≤ (8n^{2})^{ −
1} f′′_{C[0,1]}
Assignment 6  Due Friday, October 21, 2016.
 Read section 2.2.7 and the notes
on
Splines and Finite Element Spaces.
 Do the following problems.
 Section 2.2: 14 (Hint: assume that Fubini's theorem holds.)
 Compute the complex form of the Fourier series for $f(x) =
e^{2x}$, $0 \le x \le 2\pi$. Use this Fourier series and Parseval's
theorem to sum the series $\sum_{k=\infty}^\infty (4+k^2)^{1}$.
 Let $f$ be a piecewise smooth, continuous $2\pi$ periodic
function having a piecewise continuous derivative, $f'$. Show that
\[
\int_0^{2\pi} f'(x) e^{inx}dx = in\int_0^{2\pi} f(x) e^{inx}dx.
\]
(Don't forget to deal with the (possible) discontinuities in $f'$ when you
integrate by parts.) Use this to show that we may interchange sum and
derivative to obtain the Fourier series for $f'$. That is, if
$f(x)=\sum_{\inty}^{\infty} c_n e^{inx}$, then
\[ f'(x) = \frac{d}{dx} \big\{\sum_{\infty}^{\infty} c_n e^{inx}
\big\}=\sum_{\infty}^{\infty} c_n \frac{d}{dx}e^{inx} =
\sum_{\infty}^{\infty} inc_n e^{inx} \]
Use this to show that the sine/cosine form of the result is
$f'(x) = \sum_{n=1}^\infty n(a_n\cos(nx) b_n \sin(nx))$, where the
$a_n$'s and $b_n$'s are Fourier coefficients of the Fourier series for $f$.
 Use the previous problem to show that if $f$ is a piecewise
smooth, continuous, $2\pi$periodic function having a piecewise
continuous derivative $f'$, then the Fourier series for $f$ converges
uniformly to $f$. (Hint: Note that since $f'\in L^2[0,2\pi]$,
the series $\sum_{n=\infty}^\infty n^2c_n^2$ is convergent. Show
that $\sum_{n=\infty}^\infty c_n$ is convergent, then apply the
Weierstrass Mtest to obtain the result.)
 In class we showed that if $f(x) = \left\{ \begin{array}{cl} 1 &
0 \le x \le \pi, \\ 1 & \pi \le x< 0 \end{array} \right.$, then the
Fourier series for $f$ is given by $f(x)=\sum_{n\ \text{odd}}
\frac{4}{n \pi}\sin(nx) = \sum_{k=1}^\infty \frac{4}{(2k1)
\pi}\sin\big((2k1)x\big)$. Using this series, find the Fourier series
for $F(x) =x$, $x\le \pi$. (You will need to compute
just one integral.)
Assignment 7  Due Friday, October 28, 2016.
 Read sections 3.1, 3.2 and my notes
on Xray
Tomography and on Bounded
Operators & Closed Subspaces.
 Do the following problems.
 Section 2.2: 25(a,b), 26(b), 27(a)
 Let $S^{1/n}(1,0)$ be the space of piecewise linear splines, with
knots at $x_j=j/n$, and let $N_2(x)$ be the linear Bspline ("tent
function", see Keener, p. 81 or my notes on splines.)
 Let $\phi_j(x):= N_2(nx +1 j)$. Show that
$\{\phi_j(x)\}_{j=0}^n$ is a basis for $S^{1/n}(1,0)$.
 Let $S_0^{1/n}(1,0):=\{s\in S^{1/n}(1,0):s(0)=s(1)=0\}$. Show that
$S_0^{1/n}(1,0)$ is a subspace of $S^{1/n}(1,0)$ and that
$\{\phi_j(x)\}_{j=1}^{n1}$ is a basis for it.
 Let $H_0$ be the set of all $f\in C^{(0)}[0,1]$ such that
$f(0)=f(1)=0$ and that $f'$ is piecewise continuous. Show that
$\langle f,g\rangle_{H_0} :=\int_0^1f'(x)g'(x)dx$ defines a real
inner product on $H_0$.
 We want to use a Galerkin method to numerically solve the
boundary value problem (BVP): −u" = f(x), u(0) = u(1) = 0,
f ∈ C[0,1]
 Weak form of the problem. Let H_{0} be as in the
previous problem. Suppose that $v\in H_0$. Multiply both sides of
the equation above by $v$ and use integration by parts to show that
$ \langle u,v\rangle_{H_0} = \langle f,v\rangle_{L^2[0,1]}$. This
is called the ``weak'' form of the BVP.
 Conversely, suppose that u ∈ H_{0} is also in
C^{(2)}[0,1] and that u satisfies
⟨u,v⟩_{H0} = ∫_{0}^{1} f(x)
v(x) dx for all v ∈ H_{0}.
Show that u satisfies the BVP.
 Note that $S_0:=S_0^{1/n}(1,0)$ is a subspace of $H_0$ and let
$s_n\in S_0$ satisfy $\us_n\_{H_0} = \min_{s\in S_0}\u 
s\_{H_0}$; thus, $s_n$ is the leastsquares approximation to u from
∈ S_{0}. Expand $s_n$ in the basis from problem 2(b):
$s_n = \sum_{j=1}^{n1}\alpha_j\phi_j$. Use the normal equations and
part (a) above to show that the $\alpha_j$'s satisfy $G\alpha =
\beta$, where $\beta_j= \langle f,\phi_j\rangle_{L^2[0,1]}$ and $G_{kj}
=\langle \phi_j,\phi_k\rangle_{H_0}$
 Show that
$
G=\begin{pmatrix} 2n& n &0 &\cdots &0\\
n & 2n& n &0 &\cdots \\
0&n& 2n& \ddots &\ddots \\
\vdots &\cdots &\ddots &\ddots &n\\
0 &\cdots &0 &n &2n
\end{pmatrix}
$
Assignment 8  Due Friday, November 4, 2016.
 Read sections 3.3, 3.4, and my notes on Bounded
Operators & Closed Subspaces;
The projection theorem, the Riesz representation theorem, etc.
 Do the following problems.
 Section 3.2: 3(d) (Assume the appropriate
operators are closed and that λ is real.)
 Section 3.3: 2 (Assume the appropriate
operators are closed and that λ is real.)
 Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous.
 Consider the Sobolev space $H^1[0,1]$, with the inner product
$\langle f, g\rangle_{H^1} := \int_0^1 \big(f(x)\overline {g(x)} +
f('x)\overline {g'(x)}\big)dx$. For $f\in H^1$, let $Df=f'$. Show that
$D:H^1[0,1]\to L^2[0,1]$ is bounded, and that $\D\_{H^1 \to L^2}=1$.
 Let L be a bounded linear operator on Hilbert space $\mathcal
H$. Show that these two formulas for $\L\$ are equivalent:
 $\L\ = \sup \{\Lu\ : u \in {\mathcal H},\ \u\ = 1\}$
 $\L\ = \sup \{\langle Lu,v\rangle : u,v \in {\mathcal H},\
\u\=\v\=1\}$
 Let $k(x,y)$ be defined by
\[
k(x,y) = \left\{
\begin{array}{cl}
y, & 0 \le y \le x\le 1, \\
x, & x \le y \le 1.
\end{array}
\right.
\]

Let $L$ be the integral operator $L\,f = \int_0^1
k(x,y)f(y)dy$. Show that $L:C[0,1]\to C[0,1]$ is bounded and that the
norm $\L\_{C[0,1]\to C[0,1]}\le 1$. Actually, $\L\_{C[0,1]\to
C[0,1]}=1/2$. Can you show this?
 Show that $k(x,y)$ is a HilbertSchmidt
kernel and that $\L\_{L^2\to L^2} \le \sqrt{\frac{1}{6}}$.
 Finish the proof of the Projection Theorem: If for every $f\in
\mathcal H$ there is a $p\in V$ such that $\pf\=\min_{v\in
V}\vf\$ then $V$ is closed.
Assignment 9  Due Friday, November 11, 2016.
 Read sections 3.5, and my notes on Compact
Operators, and on
Closed Range Theorem.
 Do the following problems.
 Section 3.4: 2(b)

Let H_{0} be the inner product space defined
problem 3, HW7. This becomes a Hilbert space when we allow $f'$ to
be in $L^2[0,1]$.
 Recall that by
problem 5, HW3,
if we have $f\in H_0$, then $\f\_{C[0,1]} \le \f\_{H_0}$. (Assume
this holds for $f$ such that $f'$ is in $L^2[0,1]$.) Show that
$\Phi_y(f) := f(y)$ is a bounded linear functional on H_{0}.
 Show that for all $f\in H_0$ there is a function $G_y$ in $H_0$
for which $f(y) = \langle f, G_y\rangle$. ($G_y$ is called a
reproducing kernel and H_{0} is a reproducing
kernel Hilbert space. It is also a Green's function for boundary
value problem in
problem 4, HW7.)
 Let $L$ be a bounded operator on a Hilbert space $H$. Show that
the closure of the the range of $L$ satisfies $\overline{R(L)} =
N(L^\ast)^\perp$. (Hint: Follow the proof of the Fredholm
alternative, which is just the special case where $R(L)$ is closed.)
 A sequence {f_{n}} in a Hilbert space H is said to
be weakly convergent to f ∈ H if and only if lim_{ n
→ ∞} < f_{n},g> = < f,g> for every
g∈H. When this happens, we write f = wlim f_{n}. For
example, if {φ_{n}} is any orthonormal sequence, then
φ_{n} converges weakly to 0. You are given that every weakly
convergent sequence is a bounded sequence (i.e. there is a constant C
such that f_{n} ≤ C for all n). Prove the following:
Let K be a compact linear operator on a Hilbert space
H. If f_{n} converges weakly to f, then Kf_{n}
converges to Kf — that is, lim_{ n → ∞} 
Kf_{n}  Kf  = 0.
Hint: Suppose this doesn't happen, then there will be a subsequence of
{f_{n}}, say {f_{nk}}, such that 
Kf_{nk}  Kf  ≥ ε for all k. Use this
and the compactness of K to arrive at a contradiction. We remark that
the converse is also true. If a bounded linear operator $K$ maps
weakly convergent sequences into convergent sequences, then $K$ is
compact.
 Show that every compact operator on a Hilbert space is a bounded operator.
 Consider the finite rank (degenerate) kernel k(x,y) =
φ_{1}(x)ψ_{1}(y) +
φ_{2}(x)ψ_{2}(y),
where φ_{1} = 6x3, φ_{2} = 3x^{2},
ψ_{1} = 1, ψ_{2} = 8x − 6. Let Ku=
∫_{0}^{1} k(x,y)u(y)dy. Assume that L =
Iλ K has closed range,

For what values of λ does the integral equation
u(x)  λ∫_{0}^{1} k(x,y)u(y)dy =f(x)
have a solution for all f ∈ L^{2}[0,1]?
 For these values, find the solution u = (I −
λK)^{−1}f — i.e., find the resolvent.
 For the values of λ for which the equation
does not have a solution for all f, find a condition on f
that guarantees a solution exists. Will the solution be unique?
Assignment 10  Due Friday, November 18, 2016.
 Read sections 3.5, 3.6, 4.1 and my notes on
Spectral Theory for Compact Operators.
 Do the following problems.
 Section 3.4: 2(c), 6 (The condition in 6 should be
λμ_{i} ≠ 1.)
 Section 3.5: 1(b), 2(b)
 (This is a variant of problem 3.4.3 in Keener.) Consider the
operator $Ku(x) = \int_{1}^1 (1xy)u(y)dy$ and the eigenvalue
problem $\lambda u = Ku$.
 Show that $K$ is a selfadjoint, HilbertSchmidt operator.
 Let $f\in C[1,1]$. If $v= Kf$, show that $v"=2f$,
$v(1)+v(1)=0$, and $v'(1)+v'(1)$.
 Use the previous part to convert the eigenvalue problem $\lambda
u = Ku$ into this eigenvalue problem:
\[
\left\{
\begin{align}
u"+&\frac{2}{\lambda} u =0,\\
u(1)+&u(1) =0 \\
u'(1)+ &u'(1)=0.
\end{align}
\right.
\]
Solve the eigenvalue above to get the eigenvalues and
eigenvectors of $\lambda u = Ku$. Explain why the eigenvectors form
a complete set for $L^2[1,1]$.
 In the following, H is a Hilbert space and B(H) is the set of
bounded linear operators on H. Let L be in B(H) and let S = sup
{< Lu, u> : u ∈ H, u = 1}.
 Show that for all w ∈ H, < Lw, w> ≤
S w^{2} ≤ L w^{2}. (Hence, S ≤ L.)
 Verify the identity < L(u+αv), u+αv> − <
L(uαv), uαv> = 2α<
Lu,v>+2α< Lv,u>, where α = 1.
 Let L be a selfadjoint operator. Use
(a) and (b) to show that S = L. (Hint: Choose α so
that α< Lu,v> = < Lu,v>)
 Suppose that H is a complex Hilbert space. If L ∈
B(H), then, again using (a) and (b), show that
S ≤ L ≤ 2S.
 For H = C^{2}, let let $L =
\begin{pmatrix}
0& 1\\
0 & 0 \end{pmatrix}.
$
Show that S = 1/2 and L = 1.
 For H = R^{2}, which is a real Hilbert space, and
again let $L = \begin{pmatrix} 0& 1\\ 1 & 0 \end{pmatrix}. $ Show
that S = 0 and L = 1. Thus, (d) fails for real H.
 Let K be a compact, selfadjoint
operator on a Hilbert space H, and let M be closure of the the span of
the set of eigenvectors {φ_{j}} corresponding to all
eigenvalues of K such that λ_{j} ≠ 0. (Note: both M
and M^{⊥} may be infinite dimensional.)
 Show that M and M^{⊥} are both invariant under K.
 Show that K restricted to M^{⊥} is compact.
 Show that either M^{⊥} = {0} or that
M^{⊥} is the eigenspace for λ = 0.
 Show that one may choose a complete, orthonormal set for H from among
the eigenvectors of K. (Use Proposition 2.4 in
Spectral Theory for Compact Operators.)
Assignment 11  Due Friday, December 2, 2016.
 Read sections 3.6, 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1. and my notes on
Examples problems for distributions.
 Do the following problems.
 Section 4.1: 1(b), 4, 6
 Section 4.2: 1, 4, 8
 Section 4.3: 3
 Let $F:C[0,1]\to C[0,1]$ be defined by $F[u](t) :=
\int_0^1(2+st+u(s)^2)^{1}ds$, $0\le t\le 1$. Let $\ \cdot
\:=\\cdot \_{C[0,1]}$. Let $B_r:=\{u\in C[0,1]\,\, \u\\le
r\}$.
 Show that $F: B_1\to B_{1/2}\subset B_1$.
 Let $D$ be an open subset of a Banach space $V$. We say that a
map $G:D\to V$ is Lipschitz continuous on $D$ if and only
if there is a constant $0<\alpha$ such that $\G[u]G[v]\\le
\alpha \uv\$. Show that $F$ is Lipschitz continuous on $B_1$,
with Lipschitz constant $0<\alpha \le 1/2$.
 Show that $F$ has a fixed point in $B_1$.
 Let $L$ be in $\mathcal B (\mathcal H)$.
 Show that $\L^k\ \le \L\^k$, $k=2,3,\ldots$.
 Let $\lambda \L\<1$. Show that
\[
\big\(I  \lambda L)^{1}  \sum_{k=0}^{n1}\lambda^k L^k\big\ \le
\frac{\lambda^n \L\^n}{1  \lambda \L\}.
\]
 Let $L$ be as in
problem 6, HW8. Use the bound on $\L\$ in 6(b) of this problem
to estimate how many terms of the Neumann expansion it would require
to approximate $(I  \lambda L)^{1}$ to within $10^{8}$, if
$\lambda\le 0.2$.
 Let $Lu=u''$, $u(0)=0$, $u'(1)=2u(1)$.
 Show that the Green's function
for this problem is
\[
G(x,y)=\left\{
\begin{array}{rl}
(2y1)x, & 0 \le x < y \le 1\\
(2x1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
 Let $Kf(x) := \int_0^1G(x,y)f(y)dy$. Show that $K$ is a selfadjoint
HilbertSchmidt operator, and that $0$ is not an eigenvalue of $K$.
 Use (b) and the spectral theory of compact operators to show the
orthonormal set of eigenfunctions for $L$ form a complete set in
$L^2[0,1]$.
Updated 11/20/2016.