Math 641-600 — Fall 2024

• Read sections 1.1-1.4
• Do the following problems.
  1. Section 1.1: 4, 5, 7(a), 8, 9(a) (Do the first 3, but without software.)
  2. Section 1.2: 9
  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 there is a unique vector $p \in U$ that satisfies $\min_{u\in U}\|v-u\| = \|v-p\|$ if and only if $v-p\in U^\perp$.
     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 $P$ has the following properties:
        1. $P$ maps $V$ onto $U$.
        2. $P$ is a linear map.
        3. $P$ satisfies $P^2 = P$ and $P^\ast=P$.  ($P$ is called an orthogonal projection. The vector $p$ is the orthogonal projection of $v$ onto $U$.)
        4. $U^\perp= \{w\in V\colon Pw=0\}$ and $V=U\oplus U^\perp$, where $\oplus$ indicates the direct sum of the two spaces. (This and the next exercise are easy, but important.)
        5. $I -P$ is the projection of $V$ onto $U^\perp$, and that the Pythagorean theorem, $\|v\|^2=\|Pv\|^2+\|(I-P)v\|^2$, holds.

  4. Let $U$ and $V$ be as in the previous exercise. Suppose that $U$ is finite dimensional and that $B=\{v_1,v_2,\ldots,v_n\}$ is an ordered basis for $U$. In addition, let $G$ be the $n\times n$ matrix with entries $G_{jk}= \langle v_k,v_j\rangle$.
     1. Show that $G$ is positive definite and thus invertible.
     2. In part 3(a), you showed that $p\in U$ is the minimizer of $\min_{u\in U}\|v-u\|$ if and only if $v-p$ is in $U^\perp$. Use this to show the following: Let $v\in V$ and $d_k := \langle v,v_k\rangle$. Show that $p$ exists for every $v$ and, relative to the basis $B$, is given by $p=\sum_j x_j v_j\in U$, where the $x_j$'s satisfy the normal equations, $d_k = \sum_{j=1}^n G_{kj}x_j$. (In matrix form, $\mathbf x = G^{-1}\mathbf d$.) Remark: Since the normal equations don't depend on the choice of $B$, $p$ itself is independent of the choice of basis.
     3. Show that if B is orthonormal, then $p=Pv=\sum_j \langle v,u_j\rangle u_j$, so $G=I$ in this case.

• Read the notes on Banach spaces and Hilbert Spaces, and sections 2.1 and 2.2 in Keener.

• Do the following problems.

  1. Section 1.2: 3, 4, 8

  2. Section 1.3: 2, 3, 5

  3. Let $V$ be an $n$ dimensional vector space and suppose $L:V\to V$ is linear.
     1. Let $B=\{v_1,\ldots, v_n\}$ and let $A_L$ be the matrix of $L$ relative to $B$. Show that both Trace($A_L$) and $\det(A_L)$ are independent of the choice of the basis $B$.
     2. Explain why $\text{Trace}(L) := \text{Trace}(A_L)$ and $\det(L) := \det(A_L)$ are well defined.
     3. Let $L:P_2\to P_2$ be given by $L(p)= \big((1-x^2)p'\big)' + 7p$. Find $\text{Trace}(L)$ and $\det(L)$.

  4. Let $\mathcal P_n$ be the polynomials of degree $n$ or less. Let $L(u) = (1-x^2)u''-xu'$. Show that $L:\mathcal P_n \to \mathcal P_n$. For $n=3$, find the eigenvalues and eigenvectors of $L$, where the eigenvectors are given in terms of polynomials. (The polynomials you get are the Chebyshev polynomials.)

  5. 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.

  6. 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. Eigenvectors corresponding to distinct eigenvalues are orthogonal.
     4. Show that from the eigenvectors of $U$ one may select an orthonormal basis for $V$ – i.e., $U$ is diagonalizable. (Hint: follow the proof for self-adjoint case.)

Assignment 3 - Due Wednesday, September 11, 2024.

• Read Keener's sections 2.1 and the notes on Lebesgue integration.
• Do the following problems.
  1. Section 2.1: 4, 5, 6

  2. Before one can define a norm or inner product on some set, one has to show that the set is a vector space -- i.e., that linear combinations of vectors are in the space. Do this for the spaces of sequences below. The inequalities from the previous assignment will be useful.
     1. $\ell^2=\{x=\{x_n\}_{n=1}^\infty\colon \sum_{j=1}^\infty |x_j|^2\}$
     2. $\ell^p=\{x=\{x_n\}_{n=1}^\infty\colon \sum_{j=1}^\infty |x_j|^p\}$, all $1\le p<\infty$, $p\ne 2$.
     3. $\ell^\infty = \{x=\{x_n\}_{n=1}^\infty\colon \sup_j|x_j|<\infty \}$.

  3. Show that, for all $1\le p <\infty$, $\|x\|_p := \big(\sum_{j=1}^\infty |x_j|^p \big)^{1/p}$ defines a norm on $\ell^p$.

  4. Show that $\ell^2$ is an inner product space, with $\langle x,y\rangle = \sum_{j-1}^\infty x_j \bar y_j$ being the inner product, and that with this inner product it is a Hilbert space. Bonus: show that it is separable.

  5. Let $C^1[0,1]$ be the set of all continuously differentiable real-valued functions on $[0,1]$. Show that $C^1[0,1]$ is a Banach space if $\|f\|_{C^1} := \max_{x\in [0,1]}|f(x)| + \max_{x\in [0,1]}|f'(x)|$.

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

  7. $L^1$ minimization. Find the straight line $y=a+bx$ that minimizes $\int_0^1 |e^{-x} - a - bx|dx$, by following these steps.
     1. Whatever the minimizer is, geometric considerations show that $e^{-x}$ and $a+bx$ will cross at two points, 0 < s < t < 1. Find these two points by minimizing, over $a,b$, the area $A$ between the $f(x)$ and $a+bx$: \begin{align} A=\int_0^1 |e^{-x} - a - bx|dx & = \\ \int_0^s (e^{-x} - a - bx)dx + \int_s^t (a+ bx-e^{-x})dx &+ \int_t^1 (e^{-x} - a - bx)dx. \end{align}
     2. Use the crossing conditions $a+bs=e^{-s}$ and $a+bt=e^{-t}$ to find $a$ and $b$.
     3. Solve the same problem, but in $L^2[0,1]$ instead of $L^1[0,1]$. Make a plot $e^{-x}$ and both straight lines on the same axes.

• Read the notes on Lebesgue integration and on Orthonormal sets and expansions.
• Do the following problems.

  1. Section 2.2: 8(a,b,c) (FYI: the formula for $T_n(x)$ has an $n!$ missing in the numerator.), 9, 10

  2. 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 $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.

  3. 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 $L^2_w$.

  4. A measurable function whose range consists of a finite number of values is a simple function — see Lebesgue integration, p. 5. Use the definition of the Lebesgue integral in in terms of Lebesgue sums, from eqn. 2, to show that, in terms of this definition, the integral of a simple function ends up being the one in eqn. 3 on p. 6.

  5. Consider the function $f:[0,1] \to \{0,1\}$, where $f(x)=0$, if $x=1/n$, $n=1,2, 3, \ldots$ and $f(x)=1$, otherwise. Show that the Lebesgue integral of $f$ is 1. Does the Riemann integral for this function exist? Prove your answer.

Assignment 5 - Due Wednesday, September 25, 2024.

• Read sections 2.2.2-2.2.4 and the notes on Approximation of Continuous Functions.
• Do the following problems.
  1. Section 2.1: 10, 11

  2. This problem illustrates how to find the (improper) integral of an unbounded function. Consider the function $f(x)=x^{-1/2}$. Use the formula in the eqn. 5 of the notes to show that, in the Lebesgue sense, $\int_0^1 f(x)dx=2$. Do this just using $x^{-1/2}$ and its properties.

  3. 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.

  4. 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.

  5. 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. 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 $\omega(f, \delta) = \inf S_\delta$
     2. 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.)
     3. Show that $\lim_{\delta \downarrow 0} \omega(f,\delta) = 0$.

• Read sections 2.2.2-2.2.4, the notes on Fourier series, and the notes on the discrete Fourier transform.
• Do the following problems.
  1. Section 2.2: 14

  2. Let $g$ be in $C^2[a,b]$. Let $h = b-a$.
     1. Show that if g(a) = g(b) = 0, then $ \|g\|_{C[a,b]} \le (h^2/8) \|g''\|_{C[a,b]}$. Give an example that shows that $1/8$ is the best possible constant.

     2. Use the previous part to show that if $f\in C^{(2)}[0,1]$, then the equally spaced linear spline interpolant $s_f$ satisfies $\|f - s_f\|_{C[0,1]} \le (8n^2)^{-1}\|f''\|_{C[0,1]}$. If $f\in C^{(3)}[0,1]$, will $s_f$ satisfy $\|f - s_f\|_{C[0,1]} \le Cn^{-3}\|f'''\|_{C[0,1]}$? Prove the statement or give a counterexample.

  3. In proving the Weierstrass Approximation, we did the case $x-j/n>\delta$. Do the case $j/n-x>\delta$.

  4. Prove this: Let $g$ be a $2\pi$ periodic function (a.e.) that is integrable on each bounded interval in $\mathbb R$. Then, $\int_{-\pi+c}^{\pi+c} g(u)du$ is independent of $c$. In particular, this imples that $\int_{-\pi+c}^{\pi+c} g(u)du=\int_{-\pi}^\pi g(u)du=\int_0^{2\pi} g(u)du$.

  5. Compute the Fourier series for the following $2\pi$ functions. In each case sketch three periods of the function to which the series converges.
     1. $f(x) = x$, $0\le x \le 2\pi$. (sine/cosine form)
     2. $f(x) = |x|$, $-\pi \le x \le \pi$. (sine/cosine form)
     3. $f(x) = e^{2x}$, $-\pi \le x \le \pi$. (complex form)
     4. $f(x) = e^{2x}$, $0 \le x \le 2\pi$. (complex form) Does the FS you get contradict the result in 4 above? Explain your answer.

  6. Use the FS from 5(d) above and Parseval's theorem to sum the series $\sum_{k=-\infty}^\infty (4+k^2)^{-1}$.

  7. The following problem is aimed at showing that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.
     1. Show that the FS for a linear spline $s(x)$ that satisfies $s(-\pi)=s(\pi)$ is uniformly convergent to $s(x)$ on $[-\pi,\pi]$.
     2. Show that such splines are dense in $L^2[-\pi,\pi]$.
     3. Show that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.

• Read section 2.2.7 and the notes on Splines and Finite Element Spaces.
• Do the following problems.
  1. Section 2.2: 25

  2. Let $\mathcal S_n$ be the set of $n$-periodic, complex-valued sequences.

     1. Suppose that $\mathbf x \in \mathcal S_n$. Show that $ \sum_{j=m}^{m+n-1}{\mathbf x}_j = \sum_{j=0}^{n-1}{\mathbf x}_j $. (This is the DFT analogue of problem above.)

     2. Prove the Convolution Theorem for the DFT. (See the Notes on the Discrete Fourier Transform, pg. 3.)

  3. Let α, ξ, η be n-periodic sequences, and let a, x, y be column vectors with entries a₀, ..., a_n-1, etc. Show that the convolution η = α∗ξ is equivalent to the matrix equation y = Ax, where A is an n×n matrix whose first column is $\mathbf a$, and whose remaining columns are $\mathbf a$ with the entries cyclically permuted. Such matrices are called . Use the DFT and the convolution theorem to find the eigenvalues of a cyclic matrix. Use this method, along with your favorite software, to find the eigenvalues and eigenvectors of the matrix below. (For this matrix, $\mathbf a =(3\ 1\ 4\ 5)^T$.) \[ \begin{pmatrix} 3 &5 &4 &1 \\ 1 &3 &5 &4 \\ 4 &1 &3 &5\\ 5 &4 &1 &3 \end{pmatrix} \]

  4. 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.

  5. Consider the space of cubic Hermite splines $S_0^{1/n}(3,1)\subset S^{1/n}(3,1)$ that satisfy $s(0)=s(1)=0$. Show that $\langle u,v\rangle = \int_0^1 u''v''dx$ defines an inner product on $S_0^{1/n}(3,1)$.

• Read sections 3.2, 3.3, the notes on Bounded Operators & Closed Subspaces, and the notes on the projection theorem, the Riesz representation theorem, etc.
• Do the following problems.
  1. Section 2.2: 27(a)

  2. 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]. Let $H^1_0$ be the space of all functions $g:[0,1]\to \mathbb R$ such that $g'$ is in $L^2[0,1]$ and $g(0)=g(1)=0$. Define an inner product on $H^1_0$ by $ \langle f,g\rangle_{H^1_0}=\int_0^1 f'g'dx$. You are given that $H^1_0$ is a Hilbert space.
     1. Weak form of the problem. Suppose that $v\in H^1_0$. Multiply both sides of the equation $-u''=f$ by $v$ and, assuming you can use integration by parts, show that $ \langle u,v\rangle_{H^1_0} = \langle f,v\rangle_{L^2[0,1]}$. This is called the ``weak'' form of the BVP.

     2. Conversely, suppose that $u\in H^1_0$ is also in $C^2[0,1]$ and that if for all $v\in H^1_0$, $u$ satisfies \[ \langle u,v\rangle_{H^1_0} = \langle f,v\rangle_{L^2[0,1]}. \] then $-u''=f$.

     3. Let $S_0^{1/n}(1,0) \subset S^{1/n}(1,0)$ be the set of all linear splines that are $0$ at $x=0$ and $x=1$; note that $S_0^{1/n}(1,0)$ a subspace of $H^1_0$. Let $s_n\in S^{1/n}_0(1,0)$ satisfy $\|u-s_n\|_{H^1_0} = \min_{s\in S^{1/n}_0(1,0)}\|u - s\|_{H^1_0}$; thus, $s_n$ is the least-squares approximation to $u$ from $S^{1/n}_0(1,0)$. Expand $s_n$ in the basis from Assignment 7, problem 4(b): $s_n = \sum_{j=1}^{n-1}\alpha_j\phi_j$ Use the normal equations for the problem in connection with the weak form of the problem 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}. $

  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. 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 $\|L\|_{C[0,1]\to C[0,1]}\le 1$. Bonus (5 pts.): Show that $\|L\|_{C[0,1]\to C[0,1]}=1/2$.

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

• Read sections 3.1-3.3, the notes on the projection theorem, the Riesz representation theorem, etc, and the notes on an example of the Fredholm alternative and finding a resolvent .
• Do the following problems.

  1. Section 3.1: 2

  2. Section 3.2: 3(c) (Assume the appropriate operators are closed and that λ is real.)

  3. Let $V$ be a subspace of a Hilbert space $\mathcal H$. Show that $(V^\perp)^\perp= \overline{V}$, where $\overline{V}$ is the closure of $V$ in $\mathcal H$. Use this to show that $\mathcal H = \overline{V}\oplus V^\perp$.

  4. 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.
  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. Consider the Hilbert space $\mathcal H=\ell^2$ and let $S=\{x=(x_{1}\ x_{2}\ x_3\ \ldots)\in \ell^2: \sum_{n=1}^\infty (n^2+1)|x_n|^2 <1\}$. Show that $S$ is a precompact subset of $\ell^2$.

Assignment 10 - Due Wednesday, November 13, 2024.

• Read sections 3.3-3.5, and my notes on Compact Operators, on the Closed Range Theorem and my notes on Spectral Theory for Compact Operators.
• Do the following problems.
  1. Section 3.3: 1 (Assume the appropriate operators are closed and that λ is real.)

  2. Section 3.4: 2(b,c)

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

  4. 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?

  5. In the following, $\mathcal H$ is a Hilbert space, with inner product $\langle Lu,v\rangle$, $\mathcal B(\mathcal H )$ is the set of bounded linear operators on $\mathcal H$. Let $L$ be in $\mathcal B(\mathcal H )$ and let $ N:= \sup {|\langle Lu, u \rangle | : u\in \mathcal H, \|u\| = 1}$.
     1. Verify the identity $\langle L(u+\alpha v), u+\alpha v\rangle - \langle L(u-\alpha v), u-\alpha v\rangle = 2\bar \alpha \langle Lu, v\rangle + 2\alpha \langle Lv, u\rangle$, where $\alpha \in \mathbb C$.

     2. Show that $N \le \|L\|$.

     3. Let $L=L^\ast$ be a self-adjoint operator on $\mathcal H$, which may be a real or complex Hilbert space. Use (a) and (b) to show that $N= \|L\|$. (Hint: In the complex case, choose $\alpha$ so that $ \alpha \langle Lu,v\rangle = |\langle Lu,v\rangle|$. For the real case, use $\alpha=\pm 1$, as needed.)

     4. Suppose that $\mathcal H$ is a complex Hilbert space. If $ L\in \mathcal B(\mathcal H)$. Use (a) and (b) to show that $\|L\| \le 2N$.

     5. For the real Hilbert space, $\mathcal H = \mathbb R^2$, let $L = \begin{pmatrix} 0& 1\\ -1 & 0 \end{pmatrix}.\ $ Show that $\|L\| = 1$, but $N=0$.

  6. Let $L\in \mathcal B(\mathcal H)$. Suppose that for all $f\in N(L)^\perp$ there is a constant $c>0$ such that $\|Lf\|\ge c\|f\|$, where $c$ is independent of $f$. Show that $R(L)$ is closed.

• Read sections 3.5, 3.6, 4.1 and and my notes on example problems for distributions.
• Do the following problems.
  1. Section 3.4: 6 (The condition in 6 should be $\lambda \mu_i\ne 1$.)

  2. Section 3.5: 2(b)

  3. Section 3.6: 1

  4. Let $E$ be the set of all $2\pi$ periodic even functions in $L^2[-\pi,\pi]$. (Note that $f\in E$ if and only if $f$ has the series expansion $f(x) = f_0 + \sum_{n=1}^\infty f_n\cos(nx)$.) Let $k$ be in $E$, with $k$ having the cosine series $k(x) = k_0 + \sum_{n=1}^\infty k_n\cos(nx)$. Finally, the operator $Kf=\int_{-\pi}^\pi k(x-y)f(y)dy$.
     1. Show that $K$ is Hilbert-Schmidt.
     2. Show that $Kf(x)=2\pi k_0f_0+ \sum_{n=1}^\infty \pi k_nf_n \cos(nx)$.
     3. Assume that all of the $k_n$'s are nonzero, find the eigenvalues and eigenvectors. Is $0$ an eigenvalue? What is $\sigma(K)$?
     4. Show that $K$ does not have a bounded inverse.
     5. If $k(x)=\sum_{n=0}^\infty \frac{\cos(2n+1)x}{(2n+1)^2}$, what are the eigenvalues and eigenvectors for $k$? Show that $0$ is an eigenvalue of $K$ and find the eigenspace of $0$.

  5. In Proposition 2.5 in my notes on Compact Operators, show that $\|K_{n+1}\| < \|K_n\|$.

  6. Consider the kernel $k(x,y)=\min(x,y)$, $0\le x,y\le 1$.
     1. Show that $Ku=\int_0^1 k(x,y)u(y)dy$ is a compact, self-adjoint operator operator.
     2. Let $U(x)=\int_0^x u(y)dy - \int_0^1 u(y)dy$. Show that $Ku(x) = -\int_0^x U(y)dy$, and that $ \int_0^1 (Ku(x))\,u(x)dx = \int_0^1 U(x)^2dx$.
     3. Use this identity to show that $0$ is not an eigenvalue of $K$ — i.e., $N(K)=\{0\}$.
     4. Show that there is no constant $c>0$ such that $c\|u\|\le \|Ku\|$. Explain why this implies $K^{-1} \not\in \mathcal B(\mathcal H)$. (Hint: consider the sequence $u_n(x) = \sqrt{2} \cos(n\pi x)$.) The point here is that $\lambda=0$ is not an eigenvalue of $K$, but is in the spectrum of $K$.

• Read sections 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1 and my notes on and my notes on example problems for distributions.
• Do the following problems.
  1. Section 4.1: 4, 7

  2. Section 4.2: 1, 3, 4

  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. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha \le 1/2$.
     3. Does $F$ have a fixed point in $B_1$? Why or why not? Be brief.

  5. 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 $Gf(x) := \int_0^1 g(x,y)f(y)dy$. Show that $G$ is a self-adjoint Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $G$.
     3. Use (b) and the spectral theory of compact operators to show the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$.

  6. Suppose that $Lu= u''+\lambda u$, with Dom$(L)=\{u\in L^2[0,\infty):Lu\in L^2[0,\infty)\ \text{and}\ u(0)=0\}$, where $\lambda\in \mathbb C \setminus [0,\infty)$. In addition, choose $\text{Im}\sqrt{\lambda}>0$. Show that the Green's function for $L$ is given by \[ g(x,y, \lambda)=\begin{cases} \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda} xe^{i\sqrt{\lambda} y} & 0\le x\le y<\infty\\ \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda} y)e^{i\sqrt{\lambda}x} & 0\le y\le x<\infty \end{cases}\] Hint: follow the procedure used in Keener, pg. 150, to solve a similar problem.

Updated 11/24/2024.