- Read sections 1.1-1.4
- Do the following problems.
- Section 1.1: 4, 5, 7(a), 8, 9(a) (Do the first 3, but without software.)
- Section 1.2: 9
- 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 there is a unique vector $p \in U$ that satisfies $\min_{u\in U}\|v-u\| = \|v-p\|$ if and only if $v-p\in U^\perp$.
- 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. (This is easy, but important.)

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

- Read the notes
on
Banach spaces and Hilbert Spaces, and sections 2.1 and 2.2 in
Keener.
- Do the following problems.
- 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.
- Section 1.3: 2, 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?
- 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.

- Show that if α, β are positive and α + β
=1, then for all u,v ≥ 0 we have
- 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 $b=\begin{pmatrix} 1\\ 3\\ 7 \end{pmatrix}$.
- 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.)
- Let U be a unitary, n×n matrix. Show that the following hold.
- < U
**x**, U**y**> = <**x**,**y**> - The eigenvalues of U all lie on the unit circle, |λ|=1.
- Eigenvectors corresponding to distinct eigenvalues are orthogonal.

- < U

- 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.

**Assignment 3** - Due Wednesday, September 19, 2018.

- Read Keener's sections 2.1 and the notes on Lebesgue integration.
- Do the following problems.
- Section 2.1: 5
- 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.- $\ell^2=\{x=\{x_n\}_{n=1}^\infty\colon \sum_{j=1}^\infty |x_j|^2\}$
- $\ell^p=\{x=\{x_n\}_{n=1}^\infty\colon \sum_{j=1}^\infty |x_j|^p\}$, all $1\le p<\infty$, $p\ne 2$.
- $\ell^\infty = \{x=\{x_n\}_{n=1}^\infty\colon \sup_{1\lr j}|x_j|<\infty \}$.

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

- Section 2.1: 5

**Assignment 4** - Due Wednesday, September 26, 2018.

- Read the notes on Lebesgue integration and on Orthonormal sets and expansions.
- Do the following problems.
- Section 2.1: 10
- Section 2.2: 1 (Use $w=1$.), 8(a,b,c) (FYI: the formula for
$T_n(x)$ has an $n!$ missing in the numerator.), 9
- 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 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.

- Verify that the pointwise limit of f
- 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 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 $\mathcal H$.)
- Every vector in $\mathcal H$ may be uniquely represented as the series $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.

- Section 2.1: 10

Updated 9/19/2018.