General $L^{p}$ Spaces: Completeness, Convolution, and Duality

Consider the measure space $\left\{ X, \emptyset \right\}$ with $\mu (X) = \infty$. For $1 \leq p \leq \infty$, characterize the space $L^{p}(X, \mu )$.

For $1 \leq p < \infty$ and an index $n$, define $f_{n}(x) = n^{1 / p}$ if $0 \leq x \leq 1 / n$ and $f_{n}(x) = 0$ if $1 / n < x \leq 1$. Let $f$ be identically zero on $[0,1]$. Show that $\left\{ f_{n}\right\}$ converges pointwise to $f$ but does not converge in $L^{p}([0,1], m)$. Why does the Vitali $L^{p}$ Convergence Criterion not apply?

For $1 \leq p < \infty$ and an index $n$, let $f_{n}: \mathbb {R} \rightarrow \mathbb {R}$ be the characteristic function of $[n, n+1]$. Let $f$ be identically zero on $\mathbb {R}$. Show that $\left\{ f_{n}\right\}$ converges pointwise to $f$ but does not converge in $L^{p}(\mathbb {R}, m)$. Why does the Vitali $L^{p}$ Convergence Criterion not apply?

Prove the Vitali $L^{p}$ Convergence Criterion.

For a measure space $(X, \mathcal{M}, \mu )$ and $0 < p < 1$, define $L^{p}(X, \mu )$ to be the collection of measurable functions on $X$ for which $\left|f\right|^{p}$ is integrable. Show that $L^{p}(X, \mu )$ is a linear space. For $f \in L^{p}(X, \mu )$, define $\left\| f\right\|_{p}^{p} = \int_{X}\left|f\right|^{p} d \mu$. (In general, $\left\| \cdot \right\|_{p}$ is not a norm, since Minkowski's Inequality may fail.)

Let $\left\{ f_{n}\right\}$ a Cauchy sequence in $L^{\infty }(X, \mu )$. Show that there is a measurable subset $X_{0}$ of $X$ for which $\mu \left(X \sim X_{0}\right) = 0$ and for each $\epsilon > 0$, there is an index $N$ for which

Use this to show that $L^{\infty }(X, \mu )$ is complete.

Let $c$ be the counting measure on $\mathbb {N}$. For $1 \leq p \leq \infty$, show that $L^{p}(\mathbb {N}, c) = \ell^{p}$. Then show that $\ell^{p}$ is complete.

Define $f: \mathbb {R} \rightarrow \mathbb {R}$ by $f(0) = 0$ and otherwise $f(x) = \exp \left(-1 / x^{2}\right)$. Use the Taylor expansion for $\exp (x)$ at $x = 0$ to show that for each polynomial $p(x)$,

Use this to show that $f \in C^{\infty }(\mathbb {R})$, and $f^{(k)}(0) = 0$ for all $k$. For what $x \in \mathbb {R}^{n}$ does the Taylor expansion of $f$ at $x = 0$ agree with $f(x)$ ?

Use the preceding problem to verify that the function $w$ belongs to $C_{c}^{\infty }(\mathbb {R})$.

For $1 \leq p < \infty$, let $f \in L^{p}\left(\mathbb {R}^{n}\right)$. Show that if $\int_{\mathbb {R}^{n}} f(x) g(x) d x = 0$ for all $g \in C_{c}^{\infty }\left(\mathbb {R}^{n}\right)$, then $f = 0$.

For the function $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f(x) = x^{-1 / 2} \cdot \chi_{(0,1)}(x)$, show that $(f * f)(x)$ is defined for all $x$ and is finite for $x \neq 0$.

Show that if $f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is bounded and continuous, then, as $\epsilon \rightarrow 0,\left\{ f * w_{\epsilon }\right\} \rightarrow f$ pointwise, uniformly on compact sets.

Let $k \in L^{2}\left(\mathbb {R}^{n} \times \mathbb {R}^{n}\right)$, and define the integral operator $T: L^{2}\left(\mathbb {R}^{n}\right) \rightarrow L^{2}\left(\mathbb {R}^{n}\right)$ by

Show that the linear operator $T$ is properly defined, and determine a $c \geq 0$ for which

Prove the Riesz Representation Theorem for the case $p = 1$ by adapting the proof for the case $p > 1$.

For $1 \leq p \leq \infty$ and $q$ the conjugate of $p$, let $\left\{ f_{n}\right\} \rightarrow f$ in $L^{q}(X, \mu )$ and let $\left\{ g_{n}\right\} \rightarrow g$ in $L^{p}(X, \mu )$. Show that

Find a measure space ($X, \mathcal{M}, \mu$) for which the Riesz Representation Theorem does extend to the case $p = \infty$.

For $1 \leq p < \infty$, find a representation for the bounded, linear functionals on $\ell^{p}$.

Find a representation for the bounded, linear functionals on a Dirac measure space.

Find a representation for the bounded, linear functionals on $\mathbb {R}^{n}$.

Extend the Banach-Saks Theorem to $L^{2}\left(\mathbb {R}^{n}, \mu_{n}\right)$, and use this to prove that a closed, bounded, convex subset of $L^{2}\left(\mathbb {R}^{n}, \mu_{n}\right)$ is weakly sequentially compact.

Prove that a convex, continuous real-valued function on a closed, bounded, convex subset $L^{2}\left(\mathbb {R}^{n}, \mu_{n}\right)$ takes a minimum value.

Show that $L^{\infty }\left(\mathbb {R}^{n}, \mu_{n}\right)$ is not separable.

For $1 < p < \infty$, show that a weakly convergent sequence in $L^{p}(X, \mu )$ is bounded.

Let $c$ be the counting measure on the set of natural numbers $\mathbb {N}$. For $1 \leq p < \infty$ prove that a bounded sequence in $L^{p}(\mathbb {N}, c)$ converges weakly if and only if it converges pointwise.

A linear functional $\Psi : L^{p}(X, \mu ) \rightarrow \mathbb {R}$ is said to be positive provided that $\Psi (g) \geq 0$ for each non-negative function $g$ in $L^{p}(X, \mu )$. For $1 \leq p < \infty$ and $\mu \sigma$-finite, show that each bounded linear functional on $L^{p}(X, \mu )$ is the difference of bounded, positive linear functionals.

Verify (26) and (27).

Show that $\mathcal{B} \mathcal{F} \mathcal{A}(X, \mathcal{M}, \mu )$ is a linear space on which $\left\| \cdot \right\|_{\text{var }}$ is a norm. Then show that this normed linear space is complete.

For $\Phi \in \left[L^{\infty }(X, \mu )\right]^{*}$, let $\mathcal{R}(\Phi ) \in \mathcal{B} \mathcal{F} \mathcal{A}(X, \mathcal{M}, \mu )$ be the representative of $\Phi$. Prove that $\mathcal{R}:\left[L^{\infty }(X, \mu )\right]^{*} \rightarrow \mathcal{B F} \mathcal{A}(X, \mathcal{M}, \mu )$ is an isometry.