# Math 641-600 — Fall 2018

## Assignments

Assignment 1 - Due Wednesday, September 5, 2018

• 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 12, 2018.

• 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(a,b) Hint for !0(a): You may choose the norms $\| \phi_j\|$ and $\|\psi_k\|$ to be any (convenient) positive numbers.

2. Section 1.3: 2, 3
3. Find the set of biorthogonal vectors corresponding to the set $\left\{\begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix}2 \\ 1\\ 0\end{pmatrix}\right\}$.  Suppose that $\{\mathbf a_1, \mathbf a_2, \ldots, \mathbf a_n\}$ is a set of linearly independent vectors in $\mathbb R^n$. What is the corresponding set of biorthogonal vectors?

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
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.

5. Find the $QR$ factorization for the matrix $A=\begin{pmatrix} 1 & 2 & 0\\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$. Use it to solve $Ax=b$, where $b=\begin{pmatrix} 1\\ 3\\ 7 \end{pmatrix}$.

6. 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 $\mathbf x_{min} = R^{-1}Q^\ast \mathbf y$. ($Q^\ast=Q^T$, since we are dealing with real scalars.)

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. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

Assignment 3 - Due Wednesday, September 19, 2018.

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

1. Section 2.1: 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 $\|f\|_{H^1[0,1]}^2 := \int_0^1\big( |f(x)|^2 + |f'(x)|^2\big)dx$.
7. 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.

Assignment 4 - Due Wednesday, September 26, 2018.

Updated 9/19/2018.