# Math 641-600 — Fall 2017

## Assignments

Assignment 1 - Due Wednesday, September 6, 2017.

• 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 $P$ 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. (This is easy, but important.)

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. Show that $G$ is positive definite and thus invertible.
2. 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, $d_k = \sum_{j=1}^n G_{kj}x_j$.
3. Show that if B is orthonormal, then $Pv=\sum_j \langle v,u_j\rangle u_j$.

Assignment 2 - Due Wednesday, September 13, 2017.

• 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: 10
2. Section 1.3: 2, 3, 5
3. 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
uαvβ ≤ 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 xjyj| ≤ ||x||p ||y||q.
Hint: use the inequality in part (a), but with appropriate choices of the parameters. For example, u = (|xj|/||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.
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 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.

Assignment 3 - Due Wednesday, September 20, 2017.

• Read Keener's sections 2.1 and the notes on Lebesgue integration.
• Do the following problems.

1. Section 2.1: 4, 5

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. $\ell^\infty = \{x=\{x_n\}_{n=1}^\infty\colon \sup_{1\lr j}|x_j|<\infty \}$.

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

Assignment 4 - Due Friday, September 29, 2017.

Assignment 5 - Due Wednesday, October 4, 2017.

Assignment 6 - Due Wednesday, October 11, 2017.

• Do the following problems.

1. Section 2.2: 14

2. This problem is aimed at proving the Riemann-Lebesgue Lemma, using the Weierstrass Approximation Theorem (WAT).
3. Compute the Fourier series for the following functions.
1. f(x) = x,   0≤ x ≤ 2π
2. f(x) = |x|,   − π ≤ x ≤ π
3. f(x) = e2x,  − π ≤ x ≤ π (complex form).

4. Compute the complex form of the Fourier series for $f(x) = e^{2x}$, $0 \le x \le 2\pi$. Why is this different from 3(c) above? Use this Fourier series and Parseval's theorem to sum the series $\sum_{k=-\infty}^\infty (4+k^2)^{-1}$.

5. 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, $\int_{-\pi+c}^{\pi+c} g(u)du=\int_{-\pi}^\pi g(u)du$.

6. The following problem is aimed at showing that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.
1. Consider the series ∑n cn einx, where ∑n |cn| < ∞. Show that ∑n cn einx converges uniformly to a continuous function f(x) and that the series is the Fourier series for f. (It's possible for a trigonometric series to converge pointwise to a function, but not be the Fourier series for that function.)

2. Use the previous problem to show that if $f$ is a continuous, piecewise smooth $2\pi$-periodic function, then the FS for $f$ converges uniformly to $f$. (Hint: Show that if $f'\in L^2[-\pi,\pi]$, then series $\sum_{k=-\infty}^\infty k^2|c_k|^2$ is convergent.)

3. Apply this result to show that the FS for a linear spline $s(x)$, which satisfies $s(0)=s(2\pi)$, is uniformly convergent to $s(x)$. Show that such splines are dense in $L^2[-\pi,\pi]$.

4. Show that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in $L^2[-\pi,\pi]$.

Assignment 7 - Due Monday, October 30, 2017.

Assignment 8 - Due Wednesday, November 8, 2017

Assignment 9 - Due Wednesday, November 15, 2017.

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

3. Consider the Hilbert space $\mathcal H=\ell^2$ and let $S=\{x=(\ldots x_{-2}\ x_{-1}\ x_0\ x_1 \ldots)\in \ell^2: \sum_{n=-\infty}^\infty (n^2+1)|x_n|^2 <1\}$. Show that $S$ is a precompact subset of $\ell^2$.

4. A sequence {fn} in a Hilbert space H is said to be weakly convergent to f ∈ H if and only if lim n → ∞ < fn,g> = < f,g> for every g∈H. When this happens, we write f = w-lim fn. For example, if {φn} is any orthonormal sequence, then φn 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 ||fn|| ≤ C for all n). Prove the following:
Let K be a compact linear operator on a Hilbert space H. If fn converges weakly to f, then Kfn converges to Kf — that is, lim n → ∞ || Kfn - Kf || = 0.
Hint: Suppose this doesn't happen, then there will be a subsequence of {fn}, say {fnk}, such that || Kfnk - 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. 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?

6. 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 N:= sup {|< Lu, u>| : u ∈ H, ||u|| = 1}.

Assignment 10 - Due Monday, November 27, 2017.

Assignment 11 - Due Wednesday, December 6, 2017.

• 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: 4, 7
2. Section 4.2: 1, 3, 4
3. Section 4.3: 3
4. Show that the fixed point found in the Contraction Mapping Theorem is unique.
5. 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 contraction on $D$ if and only if there is a constant $0\le \alpha$<1 such that $\|G[u]-G[v]\|\le \alpha \|u-v\|$. Show that $F$ is a Lipschitz contraction on $B_1$, with Lipschitz constant $\alpha \le 1/2$.
3. Show that $F$ has a fixed point in $B_1$.

6. Let $L$ be in $\mathcal B (\mathcal H)$.
1. Let $A$ and $B$ be in $\mathcal B (\mathcal H)$. Show that $\|AB\|\le \|A\|\,\|B\|$. Use this to show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
2. Suppose that $|\lambda| \|L\|<1$. In class (11/29/17), we showed that the truncation error $E_n$ satisfies $E_n=\big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le \frac{|\lambda|^n \|L\|^n}{1 - |\lambda| \|L\|}.$ Let $L$ be as in problem 3, HW8. Use the bound on $\|L\|$ in 3(b) to estimate how many terms of the Neumann expansion would be required to approximate $(I - \lambda L)^{-1}$ to within $10^{-8}$, if $|\lambda|\le 0.2$.

7. 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 11/29/2017.