$L^{p}$ Spaces

Suppose $\mu$ is a measure. Prove that

for all $f, g \in \mathcal{L}^{\infty }(\mu )$ and all $\alpha \in \mathbf{F}$. Conclude that with the usual operations of addition and scalar multiplication of functions, $\mathcal{L}^{\infty }(\mu )$ is a vector space.

Suppose $a \geq 0, b \geq 0$, and $1 < p < \infty$. Prove that

if and only if $a^{p} = b^{p^{\prime }}$.

Suppose $a_{1}, \ldots , a_{n}$ are nonnegative numbers. Prove that

Prove Hölder's inequality (7.9) in the cases $p = 1$ and $p = \infty$.

Suppose that $(X, \mathcal{S}, \mu )$ is a measure space, $1 < p < \infty , f \in \mathcal{L}^{p}(\mu )$, and $h \in \mathcal{L}^{p^{\prime }}(\mu )$. Prove that Hölder's inequality (7.9) is an equality if and only if there exist nonnegative numbers $a$ and $b$, not both $0$, such that

for almost every $x \in X$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space, $f \in \mathcal{L}^{1}(\mu )$, and $h \in \mathcal{L}^{\infty }(\mu )$. Prove that $\left\| f h\right\|_{1} = \left\| f\right\|_{1}\left\| h\right\|_{\infty }$ if and only if

for almost every $x \in X$ such that $f(x) \neq 0$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f, h: X \rightarrow \mathbf{F}$ are $\mathcal{S}$-measurable. Prove that

for all positive numbers $p, q, r$ such that $\frac{1}{p}+\frac{1}{q} = \frac{1}{r}$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $n \in \mathbb {N}$. Prove that

for all positive numbers $p_{1}, \ldots , p_{n}$ such that $\frac{1}{p_{1}}+\frac{1}{p_{2}}+\cdots +\frac{1}{p_{n}} = 1$ and all $\mathcal{S}$-measurable functions $f_{1}, f_{2}, \ldots , f_{n}: X \rightarrow \mathbf{F}$.

Show that the formula in 7.12 holds for $p = \infty$ if $\mu$ is a $\sigma$-finite measure.

Suppose $0 < p < q \leq \infty$.

Show that $\bigcap_{p > 1} \ell^{p} \neq \ell^{1}$.

Show that $\bigcap_{p < \infty } \mathcal{L}^{p}([0,1]) \neq \mathcal{L}^{\infty }([0,1])$.

Show that $\bigcup_{p > 1} \mathcal{L}^{p}([0,1]) \neq \mathcal{L}^{1}([0,1])$.

Suppose $p, q \in (0, \infty ]$, with $p \neq q$. Prove that neither of the sets $\mathcal{L}^{p}(\mathbb {R})$ and $\mathcal{L}^{q}(\mathbb {R})$ is a subset of the other.

Show that there exists $f \in \mathcal{L}^{2}(\mathbb {R})$ such that $f \notin \mathcal{L}^{p}(\mathbb {R})$ for all $p \in (0, \infty ] \backslash \left\{ 2\right\}$.

Suppose $(X, \mathcal{S}, \mu )$ is a finite measure space. Prove that

for every $\mathcal{S}$-measurable function $f: X \rightarrow \mathbf{F}$.

Suppose $\mu$ is a measure, $0 < p \leq \infty$, and $f \in \mathcal{L}^{p}(\mu )$. Prove that for every $\varepsilon > 0$, there exists a simple function $g \in \mathcal{L}^{p}(\mu )$ such that $\left\| f-g\right\|_{p} < \varepsilon$.

Suppose $0 < p < \infty$ and $f \in \mathcal{L}^{p}(\mathbb {R})$. Prove that for every $\varepsilon > 0$, there exists a step function $g \in \mathcal{L}^{p}(\mathbb {R})$ such that $\left\| f-g\right\|_{p} < \varepsilon$.

Suppose $0 < p < \infty$ and $f \in \mathcal{L}^{p}(\mathbb {R})$. Prove that for every $\varepsilon > 0$, there exists a continuous function $g: \mathbb {R} \rightarrow \mathbf{F}$ such that $\left\| f-g\right\|_{p} < \varepsilon$ and the set $\left\{ x \in \mathbb {R}: g(x) \neq 0\right\}$ is bounded.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space, $1 < p < \infty$, and $f, g \in \mathcal{L}^{p}(\mu )$. Prove that Minkowski's inequality (7.14) is an equality if and only if there exist nonnegative numbers $a$ and $b$, not both $0$, such that

for almost every $x \in X$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f, g \in \mathcal{L}^{1}(\mu )$. Prove that

if and only if $f(x) \overline{g(x)} \geq 0$ for almost every $x \in X$.

Suppose $(X, \mathcal{S}, \mu )$ and $(Y, \mathcal{T}, \nu )$ are $\sigma$-finite measure spaces and $0 < p < \infty$. Prove that if $f \in \mathcal{L}^{p}(\mu \times \nu )$, then

and

where $[f]_{x}$ and $[f]^{y}$ are the cross sections of $f$ as defined in 5.7.

Suppose $1 \leq p < \infty$ and $f \in \mathcal{L}^{p}(\mathbb {R})$.

Suppose $1 \leq p < \infty$ and $f \in \mathcal{L}^{p}(\mathbb {R})$. Prove that

for almost every $b \in \mathbb {R}$.

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

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

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 $f \in \mathcal{L}^{p}(\mu )$.

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

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

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.

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

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

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

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

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

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

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

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 $x \in X$.

Let

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

Suppose $1 \leq p \leq 2$.

Suppose $2 \leq p < \infty$.

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 $L^{q}(\mu )$.

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

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

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).

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

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