Math 641-600 — Fall 2016

• Read sections 1.1-1.4
• Do the following problems.
  1. Section 1.1: 3(c), 5, 7(a), 8
  2. Section 1.2: 1(a,b), 8, 9, 10(a,b)
  3. 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.)
     1. Fix $v\in V$. Show that if there exists $p\in U$ such that $p$ satisfies either (i) $\min_{u\in U}\|v-u\| = \|v-p\|$ or (ii) $v-p\in U^\perp$, then it satisfies both (i) and (ii). Moreover, if $p$ exists, it is unique.
     2. 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 $U$.)
     3. 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 $U^\perp= \{w\in V\colon Pw=0\}$.
     4. Suppose that the projection $P$ exists. Show that $V=U\oplus U^\perp$, where $\oplus$ indicates the direct sum of the two spaces.

  4. 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 $G_{jk}= \langle u_k,u_j\rangle$.
     1. 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.)
     2. Show that if B is orthonormal, then $Pv=\sum_j \langle v,u_j\rangle u_j$.

• Read the notes on Adjoints & Self-adjoint Operators, and sections 2.1 and 2.2 in Keener.

• Do the following problems.
  1. Section 1.3: 2, 3
  2. Section 1.4: 4
  3. Let $A = \begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 1\\ -1 & 1 & 1\end{pmatrix}$. Find the $QR$ decomposition of $A$.
  4. 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.
     1. Use Gram-Schmidt 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.
     2. 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 $\mathbf x_{min} = R^{-1}Q^\ast \mathbf y$.

  5. 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 $\det(A)=\det(A')$.

  6. (This is a generalization of Keener's problem 1.3.5.) Let $A$ be a self-adjoint 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.)

  7. Let U be a unitary, n×n matrix. Show that the following hold.
     1. < Ux, Uy > = < x, y >
     2. The eigenvalues of U all lie on the unit circle, |λ|=1.
     3. Show that U is diagonalizable. (Hint: follow the proof for the self-adjoint 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.
  1. Section 2.1: 3, 5

  2. 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 $L:P_2\to P_2$.
     1. Relative to the inner product above, find $L^\ast$ and $\text{Null}(L^\ast)$.
     2. 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?

  3. 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.

  4. This problem concerns several important inequalities.
     1. Show that if α, β are positive and α + β =1, then for all u,v ≥ 0 we have
        u^αv^β ≤ αu + βv.
     2. Let x,y ∈ Rⁿ, and let p > 1 and define q by q^-1 = 1 - p^-1. Prove Hölder's inequality,
        |∑_j x_jy_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
     3. Let x,y ∈ Rⁿ, 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 Rⁿ. (Hint: you will need to use Hölder's inequality, along with a trick.)

  5. 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 $\|f\|_{H^1[0,1]}^2 := \int_0^1\big( |f(x)|^2 + |f'(x)|^2\big)dx$.

• Read the notes on Lebesgue integration and on Orthonormal sets and expansions.
• Do the following problems.
  1. Section 2.1: 10, 11

  2. Section 2.2: 1 (Do $w=1$.), 10

  3. 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.

  4. Let F(s) = ∫ ₀^∞ e ^{− s t} f(t)dt be the Laplace transform of f ∈ L¹([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

  5. Let f_n(x) = n^3/2 x e^{-n x}, where x ∈ [0,1] and n = 1, 2, 3, ....
     1. Verify that the pointwise limit of f_n(x) is f(x) = 0.
     2. Show that ||f_n||_C[0,1] → ∞ as n → ∞, so that f_n does not converge uniformly to 0.
     3. Find a constant C such that for all n and x fixed f_n(x) ≤ C x^−1/2, x ∈ (0,1].
     4. Use the Lebesgue dominated convergence theorem to show that
        lim _n→∞ ∫ ₀¹ f_n(x)dx = 0.

  6. 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.)
     1. $U$ is maximal in the sense that there is no non-zero vector in $\mathcal H$ that is orthogonal to $U$. (Equivalently, $U$ is not a proper subset of any other o.n. set in $\mathcal H$.)
     2. Every vector in $\mathcal H$ may be uniquely represented as the series $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.

• Read sections 2.2.2-2.2.4 and the notes on Approximation of Continuous Functions.
• Do the following problems.
  1. Section 2.2: 8, 9. Note: the formula in Problem 8(e) has an $n!$ missing in the numerator. It should be \[ T_n(x) = \frac{(-1)^n2^n n!}{(2n)!}(1-x^2)^{1/2}\frac{d^n}{dx^n}(1-x^2)^{n-1/2} \]

  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 \[ \langle f,g\rangle_w := \int_{-1}^1 \frac{f(x)\overline{g(x)}dx}{\sqrt{1 - x^2}}. \]
     1. 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 $L^2[-1,1]$.
     2. Show that if $f\in L^\infty[-1,1]$, then $\|f\|_w \le \sqrt{\pi}\|f\|_\infty$.
     3. 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 $L^2_w$.

  3. Let $\delta>0$. We define the modulus of continuity for $f\in C[0,1]$ by $\omega(f,\delta) := \sup_{\,|\,s-t\,|\,\le\, \delta,\,s,t\in [0,1]}|f(s)-f(t)|$.
     1. Explain why $\omega(f,\delta)$ exists for every $f\in C[0,1]$.
     2. 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 $\omega(f, \delta) = \inf S_\delta$
     3. 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.)
     4. Show that $\lim_{\delta \downarrow 0} \omega(f,\delta) = 0$.

  4. Calculus problem: Let g be C² on an interval [a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
     ||g||_C[a,b] ≤ (h²/8) ||g′′||_C[a,b].
     Give an example that shows that $1/8$ is the best possible constant.

  5. Use the previous problem to show that if f ∈ C²[0,1], then the equally spaced linear spline interpolant f_n satisfies
     ||f − f_n||_C[0,1] ≤ (8n²) ^{− 1} ||f′′||_C[0,1]

• Read section 2.2.7 and the notes on Splines and Finite Element Spaces.
• Do the following problems.
  1. Section 2.2: 14 (Hint: assume that Fubini's theorem holds.)

  2. 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 $\sum_{k=-\infty}^\infty (4+k^2)^{-1}$.

  3. 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 $f$.

  4. 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 $\sum_{n=-\infty}^\infty n^2|c_n|^2$ is convergent. Show that $\sum_{n=-\infty}^\infty |c_n|$ is convergent, then apply the Weierstrass M-test to obtain the result.)
  5. 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 $f(x)=\sum_{n\ \text{odd}} \frac{4}{n \pi}\sin(nx) = \sum_{k=1}^\infty \frac{4}{(2k-1) \pi}\sin\big((2k-1)x\big)$. 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 X-ray Tomography and on Bounded Operators & Closed Subspaces.
• Do the following problems.
  1. Section 2.2: 25(a,b), 26(b), 27(a)

  2. 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 B-spline ("tent function", see Keener, p. 81 or my notes on splines.)
     1. Let $\phi_j(x):= N_2(nx +1 -j)$. Show that $\{\phi_j(x)\}_{j=0}^n$ is a basis for $S^{1/n}(1,0)$.
     2. 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 $\{\phi_j(x)\}_{j=1}^{n-1}$ is a basis for it.

  3. 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 $H_0$.

  4. 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]
     1. Weak form of the problem. Let H₀ 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.

     2. Conversely, suppose that u ∈ H₀ is also in C⁽²⁾[0,1] and that u satisfies
        ⟨u,v⟩_H0 = ∫₀¹ f(x) v(x) dx for all v ∈ H₀.
        Show that u satisfies the BVP.

     3. Note that $S_0:=S_0^{1/n}(1,0)$ is a subspace of $H_0$ and let $s_n\in S_0$ satisfy $\|u-s_n\|_{H_0} = \min_{s\in S_0}\|u - s\|_{H_0}$; thus, $s_n$ is the least-squares approximation to u from ∈ S₀. Expand $s_n$ in the basis from problem 2(b): $s_n = \sum_{j=1}^{n-1}\alpha_j\phi_j$. 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 $G_{kj} =\langle \phi_j,\phi_k\rangle_{H_0}$

     4. 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} $

• 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.
  1. Section 3.2: 3(d) (Assume the appropriate operators are closed and that λ is real.)

  2. Section 3.3: 2 (Assume the appropriate operators are closed and that λ is real.)

  3. Let V be a Banach space. Show that a linear operator L:V → V is bounded if and only if L is continuous.

  4. 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 $\|D\|_{H^1 \to L^2}=1$.

  5. Let L be a bounded linear operator on Hilbert space $\mathcal H$. Show that these two formulas for $\|L\|$ are equivalent:
     1. $\|L\| = \sup \{\|Lu\| : u \in {\mathcal H},\ \|u\| = 1\}$
     2. $\|L\| = \sup \{|\langle Lu,v\rangle| : u,v \in {\mathcal H},\ \|u\|=\|v\|=1\}$

  6. Let $k(x,y)$ be defined by \[ k(x,y) = \left\{ \begin{array}{cl} y, & 0 \le y \le x\le 1, \\ x, & x \le y \le 1. \end{array} \right. \]
     1. 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?

     2. Show that $k(x,y)$ is a Hilbert-Schmidt kernel and that $\|L\|_{L^2\to L^2} \le \sqrt{\frac{1}{6}}$.

  7. Finish the proof of the Projection Theorem: If for every $f\in \mathcal H$ there is a $p\in V$ such that $\|p-f\|=\min_{v\in V}\|v-f\|$ then $V$ is closed.

• Read sections 3.5, and my notes on Compact Operators, and on Closed Range Theorem.
• Do the following problems.
  1. Section 3.4: 2(b)

  2. Let H₀ be the inner product space defined problem 3, HW7. This becomes a Hilbert space when we allow $f'$ to be in $L^2[0,1]$.
     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 H₀.
     2. 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 H₀ is a reproducing kernel Hilbert space. It is also a Green's function for boundary value problem in problem 4, HW7.)

  3. 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.)

  4. 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 = w-lim 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.

  5. Show that every compact operator on a Hilbert space is a bounded operator.

  6. Consider the finite rank (degenerate) kernel k(x,y) = φ₁(x)ψ₁(y) + φ₂(x)ψ₂(y), where φ₁ = 6x-3, φ₂ = 3x², ψ₁ = 1, ψ₂ = 8x − 6. Let Ku= ∫₀¹ k(x,y)u(y)dy. Assume that L = I-λ K has closed range,
     1. For what values of λ does the integral equation
        u(x) - λ∫₀¹ k(x,y)u(y)dy =f(x)
        have a solution for all f ∈ L²[0,1]?
     2. For these values, find the solution u = (I − λK)⁻¹f — i.e., find the resolvent.
     3. 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.
  1. Section 3.4: 2(c), 6 (The condition in 6 should be λμ_i ≠ 1.)
  2. Section 3.5: 1(b), 2(b)
  3. (This is a variant of problem 3.4.3 in Keener.) Consider the operator $Ku(x) = \int_{-1}^1 (1-|x-y|)u(y)dy$ and the eigenvalue problem $\lambda u = Ku$.
     1. Show that $K$ is a self-adjoint, Hilbert-Schmidt operator.
     2. Let $f\in C[-1,1]$. If $v= Kf$, show that $-v"=2f$, $v(1)+v(-1)=0$, and $v'(1)+v'(-1)$.
     3. 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 $L^2[-1,1]$.

  4. 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}.
     1. Show that for all w ∈ H, |< Lw, w>| ≤ S ||w||² ≤ ||L|| ||w||². (Hence, S ≤ ||L||.)
     2. Verify the identity < L(u+αv), u+αv> − < L(u-αv), u-αv> = 2α< Lu,v>+2α< Lv,u>, where |α| = 1.
     3. Let L be a self-adjoint operator. Use (a) and (b) to show that S = ||L||. (Hint: Choose α so that α< Lu,v> = |< Lu,v>|)
     4. Suppose that H is a complex Hilbert space. If L ∈ B(H), then, again using (a) and (b), show that
        S ≤ ||L|| ≤ 2S.
        
     5. For H = C², let let $L = \begin{pmatrix} 0& 1\\ 0 & 0 \end{pmatrix}. $ Show that S = 1/2 and ||L|| = 1.
     6. For H = R², 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.

  5. Let K be a compact, self-adjoint 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.)
     1. Show that M and M^⊥ are both invariant under K.
     2. Show that K restricted to M^⊥ is compact.
     3. Show that either M^⊥ = {0} or that M^⊥ is the eigenspace for λ = 0.
     4. 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.
  1. Section 4.1: 1(b), 4, 6
  2. Section 4.2: 1, 4, 8
  3. Section 4.3: 3
  4. 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 $B_r:=\{u\in C[0,1]\,|\, \|u\|\le r\}$.
     1. Show that $F: B_1\to B_{1/2}\subset B_1$.
     2. 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 $\|G[u]-G[v]\|\le \alpha \|u-v\|$. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha \le 1/2$.
     3. Show that $F$ has a fixed point in $B_1$.

  5. Let $L$ be in $\mathcal B (\mathcal H)$.
     1. Show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
     2. Let $|\lambda| \|L\|<1$. Show that \[ \big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le \frac{|\lambda|^n \|L\|^n}{1 - |\lambda| \|L\|}. \]
     3. 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 $|\lambda|\le 0.2$.

  6. Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
     1. Show that the Green's function for this problem is \[ G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right. \]
     2. Let $Kf(x) := \int_0^1G(x,y)f(y)dy$. Show that $K$ is a self-adjoint Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $K$.
     3. 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.