Math Problems Vault

Problem 3.6.1

Prove that the triples $(X, \mathcal{R}, \mu )$ given in Examples 4 and 6 at the beginning of Section 6 are, in fact, measure spaces. For the measure in Example 3, show that

\mu \left(\bigcup _{n=1}^{\infty }\left[a_{n}, b_{n}\right]\right)=\sum _{n=1}^{\infty } \mu \left(\left[a_{n}, b_{n}\right]\right)

whenever $n \neq m$ implies $\left[a_{n}, b_{n}\right] \cap \left[a_{m}, b_{m}\right]=\emptyset$ .

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Problem 3.6.2

For $0<p<1$ , we can define the $L^{p}$ -space and $\left\| \cdot \right\|_{p}$ in the same way that we did for $1 \leq p<\infty$ . Prove, by giving a suitable example, that $\left\| \cdot \right\|_{p}$ does not satisfy the triangle inequality, and hence is not a norm.

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Problem 3.6.3

Prove, as outlined in the text, that

x y \leq \frac{x^{p}}{p}+\frac{y^{q}}{q}

for $x, y \geq 0$ and Hölder conjugates $p$ and $q$ with $1<p<\infty$ .

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Problem 3.6.4

Prove that

\left\| |f+g|^{p-1}\right\| _{q}=\left(\left\| f+g\right\| _{(p-1) q}\right)^{p-1}=\left(\left\| f+g\right\| _{p}\right)^{p-1}

for $f, g \in L^{p}(\mu )$ and $1<p<\infty$ . (This equality is used in the proof of Theorem 3.19.)

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Problem 3.6.5

Prove that $L^{\infty }$ , with norm $\left\| \cdot \right\|_{\infty }$ , is a normed linear space.

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Problem 3.6.6

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

In this exercise you will investigate relations between the various $L^{p}$ -spaces.

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(a)

Let $1 \leq p<q \leq \infty$ . Consider Lebesgue measure on $\mathbb {R}^{n}$ . Construct examples to show that neither $L^{p} \subseteq L^{q}$ nor $L^{q} \subseteq L^{p}$ holds.

(b)

Next, suppose that $1 \leq p<r<q<\infty$ . Show that $L^{p} \cap L^{q} \subseteq L^{r}$ . (This is for any measure space $(X, \mathcal{R}, \mu )$ .)

(c)

Now assume that $X$ is a finite measure space, i.e., that $X$ is measurable and that $\mu (X)<\infty$ . Prove that $L^{q} \subseteq L^{p}$ for $1 \leq p<q<\infty$ , and give an example to show that this is a proper inclusion. Now prove that

\left\| f\right\| _{1} \leq \left\| f\right\| _{p} \leq \left\| f\right\| _{q} \leq \left\| f\right\| _{\infty }, \quad 1 \leq p<q<\infty

whenever $\mu$ is a probability measure.

(d)

Prove that $\ell^{p} \subseteq \ell^{q}$ for $1 \leq p<q \leq \infty$ , and give an example to show that this is a proper inclusion. Now prove that

\left\| f\right\| _{\infty } \leq \left\| f\right\| _{q} \leq \left\| f\right\| _{p} \leq \left\| f\right\| _{1}, \quad 1 \leq p<q<\infty

Problem 3.6.7

Show that for each $f \in L^{\infty },\left|f(x)\right| \leq \left\| f\right\|_{\infty }$ almost everywhere.

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Problem 3.6.8

(a)(b)(c)

Let $I$ be an interval in $\mathbb {R}$ .

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(a)

Assume that $1 \leq p<\infty$ . Prove that $C(I)$ is dense in $L^{p}(m)$ whenever $I$ is closed and bounded.

(b)

Now drop the assumption that $I$ is closed and bounded. We certainly cannot expect $C(I)$ to be dense in $L^{P}(m)$ , since it is not even contained in it. We define the "continuous functions with compact support" on $X \subseteq \mathbb {R}$ to be

C_{c}(X)=\left\{ f \mid f \in C(X) \text{ and } \overline{\left\{ x \in X \mid f(x) \neq 0\right\} } \text{ is compact }\right\}

The set $\overline{\left\{ x \in X \mid f(x) \neq 0\right\} }$ is often called the "support" of $f$ . Again, assume that $1 \leq p<\infty$ . Prove that $C_{c}(I)$ is dense in $L^{p}(m)$ . (Note that this result subsumes the result of (a).)

(c)

Prove that $C_{c}(I)$ is not dense in $L^{\infty }(m)$ .

(The results of Exercise 8 give an alternative proof of Theorem 4.7, and give a way to define $L^{p}$ that is independent of measure theory. See the paragraph following the proof of Theorem 4.7.)

Problem 3.6.9

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

The point of this exercise is to show that the norm on $L^{p}$ , for $p \neq 2$ , does not come from an inner product. We start by considering $L^{p}([-1,1])$ , with Lebesgue measure. Set $f(x)=1+x$ and $g(x)=1-x$ .

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(a)

Show that

\begin{gathered} \left\| f\right\| _{p}^{p}=\frac{2^{p+1}}{p+1}=\left\| g\right\| _{p}^{p} \\ \left\| f+g\right\| _{p}^{p}=2^{p+1} \\ \left\| f-g\right\| _{p}^{p}=\frac{2^{p+1}}{p+1} \end{gathered}

(b)

Using part (a), show that the parallelogram equality asserts that

(p+1)^{\frac{2}{p}}=3

Verify that this equality holds for $p=2$ .

(c)

Prove that the parallelogram equality does not hold for values $p \neq 2$ . Hint: Show that the function $(p+1)^{\frac{2}{p}}-3$ is a strictly decreasing function of $p \geq 1$ and thus takes on the value zero for at most one value of $p$ .

(d)

Modify the functions given in part (a) to prove the result for $L^{p}(I)$ where $I$ is any interval, bounded or unbounded, in $\mathbb {R}$ .