Math Problems Vault

Problem 8C.1

Verify that the family $\left\{ e_{k}\right\}_{k \in \mathbf{Z}}$ as defined in the third bullet point of Example 8.51 is an orthonormal family in $L^{2}((-\pi , \pi ])$ . The following formulas should help:

\begin{aligned} & (\sin x)(\cos y) = \frac{\sin (x-y)+\sin (x+y)}{2} \\ & (\sin x)(\sin y) = \frac{\cos (x-y)-\cos (x+y)}{2} \\ & (\cos x)(\cos y) = \frac{\cos (x-y)+\cos (x+y)}{2} \end{aligned}

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Remark.

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Example 8.51 (third bullet point): for $k \in \mathbb {Z}$ , define $e_{k}: (-\pi , \pi ] \rightarrow \mathbb {R}$ by

e_{k}(t) = \begin{cases} \frac{1}{\sqrt{\pi }} \sin (kt) & \text{if } k > 0, \\ \frac{1}{\sqrt{2\pi }} & \text{if } k = 0, \\ \frac{1}{\sqrt{\pi }} \cos (kt) & \text{if } k < 0.\end{cases}

Problem 8C.2

Suppose $\left\{ a_{k}\right\}_{k \in \Gamma }$ is a family in $\mathbb {R}$ and $a_{k} \geq 0$ for each $k \in \Gamma$ . Prove the unordered sum $\sum_{k \in \Gamma } a_{k}$ converges if and only if

\sup \left\{ \sum _{j \in \Omega } a_{j}: \Omega \text{ is a finite subset of } \Gamma \right\} < \infty .

Furthermore, prove that if $\sum_{k \in \Gamma } a_{k}$ converges then it equals the supremum above.

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Problem 8C.3

Suppose $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in an inner product space $V$ . Prove that if $f \in V$ , then $\left\{ k \in \Gamma : \left\langle f, e_{k} \right\rangle \neq 0\right\}$ is a countable set.

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Problem 8C.4

Suppose $\left\{ f_{k}\right\}_{k \in \Gamma }$ and $\left\{ g_{k}\right\}_{k \in \Gamma }$ are families in a normed vector space such that $\sum_{k \in \Gamma } f_{k}$ and $\sum_{k \in \Gamma } g_{k}$ converge. Prove that $\sum_{k \in \Gamma }\left(f_{k}+g_{k}\right)$ converges and

\sum _{k \in \Gamma }\left(f_{k}+g_{k}\right) = \sum _{k \in \Gamma } f_{k}+\sum _{k \in \Gamma } g_{k}.

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Problem 8C.5

Suppose $\left\{ f_{k}\right\}_{k \in \Gamma }$ is a family in a normed vector space such that $\sum_{k \in \Gamma } f_{k}$ converges. Prove that if $c \in \mathbf{F}$ , then $\sum_{k \in \Gamma }\left(c f_{k}\right)$ converges and

\sum _{k \in \Gamma }\left(c f_{k}\right) = c \sum _{k \in \Gamma } f_{k}.

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Problem 8C.6

Suppose $\left\{ a_{k}\right\}_{k \in \Gamma }$ is a family in $\mathbb {R}$ . Prove that the unordered sum $\sum_{k \in \Gamma } a_{k}$ converges if and only if $\sum_{k \in \Gamma }\left|a_{k}\right| < \infty$ .

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Problem 8C.7

Suppose $\left\{ f_{k}\right\}_{k \in \mathbb {N}}$ is a family in a normed vector space $V$ and $f \in V$ . Prove that the unordered sum $\sum_{k \in \mathbb {N}} f_{k}$ equals $f$ if and only if the usual ordered sum $\sum_{k = 1}^{\infty } f_{p(k)}$ equals $f$ for every injective and surjective function $p: \mathbb {N} \rightarrow \mathbb {N}$ .

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Problem 8C.8

Explain why 8.58 implies that if $\Gamma$ is a finite set and $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in a Hilbert space $V$ , then $\operatorname {span}\left\{ e_{k}\right\}_{k \in \Gamma }$ is a closed subspace of $V$ .

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Problem 8C.9

Suppose $V$ is an infinite-dimensional Hilbert space. Prove that there does not exist a basis of $V$ that is an orthonormal family.

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Problem 8C.10

(a)(b)(c)(d)

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(a)

Show that the orthonormal family given in the first bullet point of Example 8.51 is an orthonormal basis of $\ell^{2}$ .

(b)

Show that the orthonormal family given in the second bullet point of Example 8.51 is an orthonormal basis of $\ell^{2}(\Gamma )$ .

(c)

Show that the orthonormal family given in the fourth bullet point of Example 8.51 is not an orthonormal basis of $L^{2}([0,1))$ .

(d)

Show that the orthonormal family given in the fifth bullet point of Example 8.51 is not an orthonormal basis of $L^{2}(\mathbb {R})$ .

Remark.

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Example 8.51:

(a)

First bullet point: for $k \in \mathbb {N}$ , let $e_{k} \in \ell^{2}$ be the sequence all of whose coordinates are $0$ except for the $k$ th coordinate, which is $1$ . Then $\left\{ e_{k}\right\}_{k \in \mathbb {N}}$ is an orthonormal family in $\ell^{2}$ .

(b)

Second bullet point: for a nonempty set $\Gamma$ , with $\ell^{2}(\Gamma ) := L^{2}(\mu )$ for $\mu$ counting measure on $\Gamma$ , define $e_{k}: \Gamma \rightarrow \mathbf{F}$ for $k \in \Gamma$ by $e_{k}(j) = 1$ if $j = k$ and $e_{k}(j) = 0$ if $j \neq k$ . Then $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in $\ell^{2}(\Gamma )$ .

(c)

Fourth bullet point: for $k$ a nonnegative integer, define $e_{k}: [0,1) \rightarrow \mathbf{F}$ (a Rademacher function) by $e_{k}(x) = 1$ if $x \in \left[\frac{n-1}{2^{k}}, \frac{n}{2^{k}}\right]$ for some odd integer $n$ , and $e_{k}(x) = -1$ if $x$ is in such an interval for some even integer $n$ . Then $\left\{ e_{k}\right\}_{k \in \left\{ 0,1,\ldots \right\} }$ is an orthonormal family in $L^{2}([0,1))$ .

(d)

Fifth bullet point: for $k$ a nonnegative integer and $m \in \mathbb {Z}$ , define $e_{k,m}: \mathbb {R} \rightarrow \mathbf{F}$ by translating the function from the fourth bullet point by $m$ : $e_{k,m}(x) = 1$ if $x \in m + \left[\frac{n-1}{2^{k}}, \frac{n}{2^{k}}\right]$ for some odd integer $n \in [1, 2^{k}]$ , $e_{k,m}(x) = -1$ if $x$ is in such a translated interval for some even integer $n \in [1, 2^{k}]$ , and $e_{k,m}(x) = 0$ if $x \notin [m, m+1)$ . Then $\left\{ e_{k,m}\right\}_{(k,m) \in \left\{ 0,1,\ldots \right\} \times \mathbb {Z}}$ is an orthonormal family in $L^{2}(\mathbb {R})$ .

Problem 8C.11

Suppose $\mu$ is a $\sigma$ -finite measure on $(X, \mathcal{S})$ and $\nu$ is a $\sigma$ -finite measure on $(Y, \mathcal{T})$ . Suppose also that $\left\{ e_{j}\right\}_{j \in \Omega }$ is an orthonormal basis of $L^{2}(\mu )$ and $\left\{ f_{k}\right\}_{k \in \Gamma }$ is an orthonormal basis of $L^{2}(\nu )$ for some countable set $\Gamma$ . For $j \in \Omega$ and $k \in \Gamma$ , define $g_{j, k}: X \times Y \rightarrow \mathbf{F}$ by

g_{j, k}(x, y) = e_{j}(x) f_{k}(y)

Prove that $\left\{ g_{j, k}\right\}_{j \in \Omega , k \in \Gamma }$ is an orthonormal basis of $L^{2}(\mu \times \nu )$ .

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Problem 8C.12

Prove the converse of Parseval's identity. More specifically, prove that if $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in a Hilbert space $V$ and

\left\| f\right\| ^{2} = \sum _{k \in \Gamma }\left| \left\langle f, e_{k} \right\rangle \right|^{2}

for every $f \in V$ , then $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal basis of $V$ .

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Problem 8C.13

(a)(b)(c)

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(a)

Show that the Hilbert space $L^{2}([0,1])$ is separable.

(b)

Show that the Hilbert space $L^{2}(\mathbb {R})$ is separable.

(c)

Show that the Banach space $\ell^{\infty }$ is not separable.

Problem 8C.14

Prove that every subspace of a separable normed vector space is separable.

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Problem 8C.15

Suppose $V$ is an infinite-dimensional Hilbert space. Prove that there does not exist a translation invariant measure on the Borel subsets of $V$ that assigns positive but finite measure to each open ball in $V$ .

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Remark.

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A subset of $V$ is called a Borel set if it is in the smallest $\sigma$ -algebra containing all the open subsets of $V$ . A measure $\mu$ on the Borel subsets of $V$ is called translation invariant if $\mu (f+E) = \mu (E)$ for every $f \in V$ and every Borel set $E$ of $V$ .

Problem 8C.16

Find the polynomial $g$ of degree at most $4$ that minimizes $\int_{0}^{1}\left|x^{5}-g(x)\right|^{2} d x$ .

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Problem 8C.17

Prove that each orthonormal family in a Hilbert space can be extended to an orthonormal basis of the Hilbert space. Specifically, suppose $\left\{ e_{j}\right\}_{j \in \Omega }$ is an orthonormal family in a Hilbert space $V$ . Prove that there exists a set $\Gamma$ containing $\Omega$ and an orthonormal basis $\left\{ f_{k}\right\}_{k \in \Gamma }$ of $V$ such that $f_{j} = e_{j}$ for every $j \in \Omega$ .

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Problem 8C.18

Prove that every vector space has a basis.

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Problem 8C.19

Find the polynomial $g$ of degree at most 4 such that

f\left(\frac{1}{2}\right) = \int _{0}^{1} f g

for every polynomial $f$ of degree at most 4.

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Problem 8C.20

(a)(b)

Suppose $G$ is a nonempty open subset of C. The Bergman space $L_{a}^{2}(G)$ is defined to be the set of analytic functions $f: G \rightarrow \mathbb {C}$ such that

\int _{G}\left|f\right|^{2} d \lambda _{2} < \infty

where $\lambda_{2}$ is the usual Lebesgue measure on $\mathbb {R}^{2}$ , which is identified with $\mathbb {C}$ . For $f, h \in L_{a}^{2}(G)$ , define $\left\langle f, h \right\rangle$ to be $\int_{G} f \overline{h} d \lambda_{2}$ .

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(a)

Show that $L_{a}^{2}(G)$ is a Hilbert space.

(b)

Show that if $w \in G$ , then $f \mapsto f(w)$ is a bounded linear functional on $L_{a}^{2}(G)$ .

Problem 8C.21

(a)(b)(c)

Let $\mathbf{D}$ denote the open unit disk in $\mathbb {C}$ ; thus

\mathbf{D} = \left\{ z \in \mathbb {C}:\left|z\right| < 1\right\}

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(a)

Find an orthonormal basis of $L_{a}^{2}(\mathbf{D})$ .

(b)

Suppose $f \in L_{a}^{2}(\mathbf{D})$ has Taylor series

f(z) = \sum _{k = 0}^{\infty } a_{k} z^{k}

for $z \in \mathbf{D}$ . Find a formula for $\left\| f\right\|$ in terms of $a_{0}, a_{1}, a_{2}, \ldots$ .

(c)

Suppose $w \in \mathbf{D}$ . By the previous exercise and the Riesz Representation Theorem (8.47 and 8.76), there exists $\Gamma_{w} \in L_{a}^{2}(\mathbf{D})$ such that

f(w) = \left\langle f, \Gamma _{w} \right\rangle \text{ for all } f \in L_{a}^{2}(\mathbf{D})

Find an explicit formula for $\Gamma_{w}$ .

Problem 8C.22

(a)(b)

Suppose $G$ is the annulus defined by

G = \left\{ z \in \mathbb {C}: 1 < \left|z\right| < 2\right\}

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(a)

Find an orthonormal basis of $L_{a}^{2}(G)$ .

(b)

Suppose $f \in L_{a}^{2}(G)$ has Laurent series

f(z) = \sum _{k = -\infty }^{\infty } a_{k} z^{k}

for $z \in G$ . Find a formula for $\left\| f\right\|$ in terms of $\ldots , a_{-1}, a_{0}, a_{1}, \ldots$ .

Problem 8C.23

Prove that if $f \in L_{a}^{2}(\mathbf{D} \backslash \left\{ 0\right\} )$ , then $f$ has a removable singularity at 0 (meaning that $f$ can be extended to a function that is analytic on $\mathbf{D}$ ).

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Problem 8C.24

The Dirichlet space $\mathcal{D}$ is defined to be the set of analytic functions $f: \mathbf{D} \rightarrow \mathbb {C}$ such that

\int _{\mathbf{D}}\left|f^{\prime }\right|^{2} d \lambda _{2} < \infty

For $f, g \in \mathcal{D}$ , define $\left\langle f, g \right\rangle$ to be $f(0) \overline{g(0)}+\int_{\mathbf{D}} f^{\prime } \overline{g^{\prime }} d \lambda_{2}$ .

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(a)

Show that $\mathcal{D}$ is a Hilbert space.

(b)

Show that if $w \in \mathbf{D}$ , then $f \mapsto f(w)$ is a bounded linear functional on $\mathcal{D}$ .

(c)

Find an orthonormal basis of $\mathcal{D}$ .

(d)

Suppose $f \in \mathcal{D}$ has Taylor series

f(z) = \sum _{k = 0}^{\infty } a_{k} z^{k}

for $z \in \mathbf{D}$ . Find a formula for $\left\| f\right\|$ in terms of $a_{0}, a_{1}, a_{2}, \ldots$

(e)

Suppose $w \in \mathbf{D}$ . Find an explicit formula for $\Gamma_{w} \in \mathcal{D}$ such that

f(w) = \left\langle f, \Gamma _{w} \right\rangle \text{ for all } f \in \mathcal{D}

Problem 8C.25

(a)(b)

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(a)

Prove that the Dirichlet space $\mathcal{D}$ is contained in the Bergman space $L_{a}^{2}(\mathbf{D})$ .

(b)

Prove that there exists a function $f \in L_{a}^{2}(\mathbf{D})$ such that $f$ is uniformly continuous on $\mathbf{D}$ and $f \notin \mathcal{D}$ .