Math Problems Vault

Problem 7B.1

Suppose $n > 1$ and $0 < p < 1$ . Prove that if $\left\| \cdot \right\|$ is defined on $\mathbf{F}^{n}$ by

\left\| \left(a_{1}, \ldots , a_{n}\right)\right\| = \left(\left|a_{1}\right|^{p}+\cdots +\left|a_{n}\right|^{p}\right)^{1 / p}

then $\left\| \cdot \right\|$ is not a norm on $\mathbf{F}^{n}$ .

?

Problem 7B.2

(a)(b)

?

(a)

Suppose $1 \leq p < \infty$ . Prove that there is a countable subset of $\ell^{p}$ whose closure equals $\ell^{p}$ .

(b)

Prove that there does not exist a countable subset of $\ell^{\infty }$ whose closure equals $\ell^{\infty }$ .

Problem 7B.3

(a)(b)

?

(a)

Suppose $1 \leq p < \infty$ . Prove that there is a countable subset of $L^{p}(\mathbb {R})$ whose closure equals $L^{p}(\mathbb {R})$ .

(b)

Prove that there does not exist a countable subset of $L^{\infty }(\mathbb {R})$ whose closure equals $L^{\infty }(\mathbb {R})$ .

Problem 7B.4

Suppose $(X, \mathcal{S}, \mu )$ is a $\sigma$ -finite measure space and $1 \leq p \leq \infty$ . Prove that if $f: X \rightarrow \mathbf{F}$ is an $\mathcal{S}$ -measurable function such that $\overline{f h} \in \mathcal{L}^{1}(\mu )$ for every $h \in \mathcal{L}^{p^{\prime }}(\mu )$ , then $f \in \mathcal{L}^{p}(\mu )$ .

?

Problem 7B.5

(a)(b)(c)

?

(a)

Prove that if $\mu$ is a measure, $1 < p < \infty$ , and $f, g \in L^{p}(\mu )$ are such that

\left\| f\right\| _{p} = \left\| g\right\| _{p} = \left\| \frac{f+g}{2}\right\| _{p}

then $f = g$ .

(b)

Give an example to show that the result in part (a) can fail if $p = 1$ .

(c)

Give an example to show that the result in part (a) can fail if $p = \infty$ .

Problem 7B.6

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $0 < p < 1$ . Show that

\left\| f+g\right\| _{p}^{p} \leq \left\| f\right\| _{p}^{p}+\left\| g\right\| _{p}^{p}

for all $\mathcal{S}$ -measurable functions $f, g: X \rightarrow \mathbf{F}$ .

?

Problem 7B.7

Prove that $L^{p}(\mu )$ , with addition and scalar multiplication as defined in 7.16 and norm defined as in 7.17, is a normed vector space. In other words, prove 7.18.

?

Problem 7B.8

Prove 7.20 for the case $p = \infty$ .

?

Problem 7B.9

Prove that 7.20 also holds for $p \in (0,1)$ .

?

Problem 7B.10

Prove that 7.23 also holds for $p \in (0,1)$ .

?

Problem 7B.11

Suppose $1 \leq p \leq \infty$ . Prove that

A_p := \left\{ \left(a_{1}, a_{2}, \ldots \right) \in \ell ^{p}: a_{k} \neq 0 \text{ for every } k \in \mathbb {N}\right\}

is not an open subset of $\ell^{p}$ .

?

Problem 7B.12

Show that there exists a sequence $f_{1}, f_{2}, \ldots$ of functions in $\mathcal{L}^{1}([0,1])$ such that $\lim_{k \rightarrow \infty }\left\| f_{k}\right\|_{1} = 0$ but

\sup \left\{ f_{k}(x): k \in \mathbb {N}\right\} = \infty

for every $x \in [0,1]$ .

?

Remark.

?

This exercise shows that the conclusion of 7.23 cannot be improved to conclude that $\lim_{k \rightarrow \infty } f_{k}(x) = f(x)$ for almost every $x \in X$ .

Problem 7B.13

Suppose $(X, \mathcal{S}, \mu )$ is a measure space, $1 \leq p \leq \infty , f \in \mathcal{L}^{p}(\mu )$ , and $f_{1}, f_{2}, \ldots$ is a sequence in $\mathcal{L}^{p}(\mu )$ such that $\lim_{k \rightarrow \infty }\left\| f_{k}-f\right\|_{p} = 0$ . Show that if $g: X \rightarrow \mathbf{F}$ is a function such that $\lim_{k \rightarrow \infty } f_{k}(x) = g(x)$ for almost every $x \in X$ , then $f(x) = g(x)$ for almost every $x \in X$ .

?

Problem 7B.14

(a)(b)

?

(a)

Give an example of a measure $\mu$ such that 7.25 fails for $p = 1$ .

(b)

Show that if $\mu$ is a $\sigma$ -finite measure, then 7.25 holds for $p = 1$ .

Problem 7B.15

(a)(b)

Let

c_{0} = \left\{ \left(a_{1}, a_{2}, \ldots \right) \in \ell ^{\infty }: \lim _{k \rightarrow \infty } a_{k} = 0\right\}

Give $c_{0}$ the norm that it inherits as a subspace of $\ell^{\infty }$ .

?

(a)

Prove that $c_{0}$ is a Banach space.

(b)

Prove that the dual space of $c_{0}$ can be identified with $\ell^{1}$ .

Problem 7B.16

(a)(b)

Suppose $1 \leq p \leq 2$ .

?

(a)

Prove that if $w, z \in \mathbb {C}$ , then

\frac{\left|w+z\right|^{p}+\left|w-z\right|^{p}}{2} \leq \left|w\right|^{p}+\left|z\right|^{p} \leq \frac{\left|w+z\right|^{p}+\left|w-z\right|^{p}}{2^{p-1}}

(b)

Prove that if $\mu$ is a measure and $f, g \in \mathcal{L}^{p}(\mu )$ , then

\frac{\left\| f+g\right\| _{p}^{p}+\left\| f-g\right\| _{p}^{p}}{2} \leq \left\| f\right\| _{p}^{p}+\left\| g\right\| _{p}^{p} \leq \frac{\left\| f+g\right\| _{p}^{p}+\left\| f-g\right\| _{p}^{p}}{2^{p-1}}

Problem 7B.17

(a)(b)

Suppose $2 \leq p < \infty$ .

?

(a)

Prove that if $w, z \in \mathbb {C}$ , then

\frac{\left|w+z\right|^{p}+\left|w-z\right|^{p}}{2^{p-1}} \leq \left|w\right|^{p}+\left|z\right|^{p} \leq \frac{\left|w+z\right|^{p}+\left|w-z\right|^{p}}{2}

(b)

Prove that if $\mu$ is a measure and $f, g \in \mathcal{L}^{p}(\mu )$ , then

\frac{\left\| f+g\right\| _{p}^{p}+\left\| f-g\right\| _{p}^{p}}{2^{p-1}} \leq \left\| f\right\| _{p}^{p}+\left\| g\right\| _{p}^{p} \leq \frac{\left\| f+g\right\| _{p}^{p}+\left\| f-g\right\| _{p}^{p}}{2}

Remark.

?

The inequalities in the two previous exercises are called Clarkson's inequalities. They were discovered by James Clarkson in 1936.

Problem 7B.18

Suppose $(X, \mathcal{S}, \mu )$ is a measure space, $1 \leq p, q \leq \infty$ , and $h: X \rightarrow \mathbf{F}$ is an $\mathcal{S}$ -measurable function such that $h f \in L^{q}(\mu )$ for every $f \in L^{p}(\mu )$ . Prove that $f \mapsto h f$ is a continuous linear map from $L^{p}(\mu )$ to $L^{q}(\mu )$ .

?

Problem 7B.19

Prove that if $1 < p < \infty$ , then $\ell^{p}$ is reflexive.

?

Remark.

?

A Banach space is called reflexive if the canonical isometry of the Banach space into its double dual space is surjective (see Exercise 20 in Section 6D for the definitions of the double dual space and the canonical isometry).

Problem 7B.20

Prove that $\ell^{1}$ is not reflexive.

?

Remark.

?

See the definition before Exercise 19 in this section for the meaning of reflexive.

Problem 7B.21

Show that with the natural identifications, the canonical isometry of $c_{0}$ into its double dual space is the inclusion map of $c_{0}$ into $\ell^{\infty }$ (see Exercise 15 for the definition of $c_{0}$ and an identification of its dual space).

?

Problem 7B.22

Suppose $1 \leq p < \infty$ and $V, W$ are Banach spaces. Show that $V \times W$ is a Banach space if the norm on $V \times W$ is defined by

\left\| (f, g)\right\| = \left(\left\| f\right\| ^{p}+\left\| g\right\| ^{p}\right)^{1 / p}

for $f \in V$ and $g \in W$ .

?