The $L^{p}$ Spaces: Duality, Weak Convergence, and Minimization

For $1 \leq p \leq \infty$, define $\Phi : L^{p}[0,1] \rightarrow \mathbb {R}$ by $\Phi (f) = \int_{0}^{1} x f(x) d x$. Show that $\Phi \in \left[L^{p}[0,1]\right]^{*}$ and determine $\left\| \Phi \right\|_{*}$.

For $x_{0} \in [a, b]$, define the functional $\Phi : C[a, b] \rightarrow \mathbb {R}$ by $\Phi (f) = f\left(x_{0}\right)$ for $f \in C[a, b]$. Prove that $\Phi \in [C[a, b]]^{*}$ and determine $\left\| \Phi \right\|_{*}$.

Define the functional $\Phi : C[a, b] \rightarrow \mathbb {R}$ by $\Phi (f) = \int_{a}^{b} f(x) d x$ for $f \in C[a, b]$. Prove that $\Phi \in [C[a, b]]^{*}$ and determine $\left\| \Phi \right\|_{*}$.

Show that a linear functional $\Phi : X \rightarrow \mathbb {R}$ is bounded if and only if there is an $r > 0$ and $x_{0} \in X$ such that the restriction $\Phi$ to $\left\{ x \in X \mid \left\| x-x_{0}\right\| \leq r\right\}$ is bounded.

Show that a sequence $\left\{ \Phi_{n}\right\}$ in $X^{*}$ fails to be bounded if and only if for each index $k, u \in X$, and $r > 0$, there is an index $n_{k}$ and $v \in X$ such that

For $f \in L^{p}(E), 1 \leq p < \infty$, show that $f = 0$ if $\Phi (f) = 0$ for all $\Phi \in \left[L^{p}(E)\right]^{*}$.

Verify (7).

Let $\mathcal{S}$ be a dense subset of $X$. Show that if $\Phi \in X^{*}$, then $\left\| \Phi \right\|_{*} = \sup \left\{ \Phi (f) \mid f \in \mathcal{S},\left\| f\right\| \leq 1\right\}$.

For $g \in L^{p}(E), 1 \leq p < \infty$, show that $g = 0$ if $\int_{A} g d m = 0$ for all $A \subseteq E$ with $m(A) < \infty$.

For $g \in C[a, b]$, show that $g = 0$ if $\int_{a}^{x} g(t) d t = 0$ for all $x \in [a, b]$.

Provide an example of a sequence of functionals in $X^{*}$ that converges pointwise on $X$ to a functional in $X^{*}$, but not as a sequence in the normed linear space $X^{*}$.

Prove Proposition 2.

Proposition 2: Let $X$ be a normed linear space. Then the collection of bounded linear functionals on $X$ is a linear space, and $\Phi \mapsto \left\| \Phi \right\|_{*}$ is a norm on this space. This normed linear space is called the dual space of $X$ and denoted by $X^{*}$.

Prove the assertion in the proof of Corollary 6 regarding the convergence of the series $\sum_{j = 1}^{\infty }[f(j)]^{2}$.

Corollary 6: Every bounded sequence $\left\{ s_{n}\right\}$ in the space $\ell^{2}$ has a subsequence that converges componentwise to a member of $\ell^{2}$. In the proof, for each $n$ one defines $f_{n}: \mathbb {N} \rightarrow \mathbb {R}$ by letting $f_{n}(j)$ be the $j$-th term of $s_{n}$; boundedness of $\left\{ s_{n}\right\}$ in $\ell^{2}$ makes $\left\{ f_{n}\right\}$ pointwise bounded on $\mathbb {N}$, so a subsequence $\left\{ f_{n_{k}}\right\}$ converges pointwise to some $f: \mathbb {N} \rightarrow \mathbb {R}$, giving componentwise convergence of $\left\{ s_{n_{k}}\right\}$ to the sequence $s$ whose $j$-th term is $f(j)$. It remains to show that the boundedness of $\left\{ s_{n}\right\}$ in $\ell^{2}$ implies the convergence of $\sum_{j=1}^{\infty } [f(j)]^{2}$, so that $s \in \ell^{2}$ — this is the assertion to be proved.

Extend Corollary 6 from $\ell^{2}$ to $\ell^{p}$, for $1 \leq p < \infty$.

Corollary 6: Every bounded sequence $\left\{ s_{n}\right\}$ in the space $\ell^{2}$ has a subsequence that converges componentwise to a member of $\ell^{2}$.

For $X$ a normed linear space, show that its dual space $X^{*}$ is a complete normed linear space.

A general functional $F: X \rightarrow \mathbb {R}$ on a normed linear space $X$ is said to be Lipschitz provided that there is a $c \geq 0$ such that $\left|F(u)-F(v)\right| \leq c\left\| u-v\right\|$ for all $u, v \in X$. The infimum of such $c$ 's is called the Lipschitz constant for $F$. Show that a linear functional $\Phi$ is bounded if and only if it is Lipschitz, in which case its Lipschitz constant is $\left\| \Phi \right\|_{*}$.

Prove that a general functional $F: X \rightarrow \mathbb {R}$ is continuous if whenever a rapidly Cauchy sequence $\left\{ u_{n}\right\}$ converges in $X$ to $u \in X$, then $\left\{ F\left(x_{n}\right)\right\} \rightarrow F(u)$.

Let $\varphi : \mathbb {R} \rightarrow \mathbb {R}$ be continuous and there is a $c \geq 0$ for which $\left|\varphi (s)\right| \leq c+c \cdot \left|s\right|^{2}$ for all $s \in \mathbb {R}$. Define $F(f) = \int_{E} \varphi \circ f d m$ for $f \in L^{2}[a, b]$. Show that $F: L^{2}[a, b] \rightarrow \mathbb {R}$ is continuous.

Define the functional $F: L^{2}[a, b] \rightarrow \mathbb {R}$ by $F(f) = \int_{[a, b]} \varphi \circ f d m$ for $f \in L^{2}[a, b]$. For the following choices of $\varphi$, show that $F$ is continuous:

Establish the Riesz Representation Theorem in the case $p = 1$ by first showing, in the notation of the proof of the theorem, that the function $\eta$ is Lipschitz and therefore it is absolutely continuous. Then follow the proof for $p > 1$.

For $1 \leq p < \infty$, by mapping a sequence $\left\{ a_{k}\right\}$ in $\ell^{p}$ to a measurable function in $L^{p}[1, \infty )$ that takes the value $a_{k}$ on each $[k, k+1)$, extend Hölder's Inequality to the $\ell^{p}$ spaces. Use this to state and prove an $\ell^{p}$ version of Lemma 9. Lemma 9: For $1 \leq p < \infty$ and $g \in L^{1}(E)$, assume that there is an $M \geq 0$ for which $\int_{E} \varphi \cdot g \, dm \leq M \cdot \left\| \varphi \right\|_{p}$ for every finitely supported, simple function $\varphi$. Then $g \in L^{q}(E)$, where $q$ is the conjugate of $p$. Moreover, $\left\| g\right\|_{q} \leq M$. Then state and prove a representation theorem for the dual of $\ell^{p}, 1 \leq p < \infty$.

Show that if every bounded linear functional on $L^{\infty }[a, b]$ were represented by a function in $L^{1}[a, b]$, then the dual of $L^{\infty }[a, b]$ would be separable.

Let $\Phi \in [C[a, b]]^{*}$. For $x \in [a, b]$, define $g_{x}:[a, b] \rightarrow \mathbb {R}$ by $g(t) = a-t$ for $a \leq t \leq x$ and $g_{x}(t) = a-x$ for $x \leq t \leq b$. Define $\eta (x) = \Phi \left(g_{x}\right)$. Show that $\eta :[a, b] \mapsto \mathbb {R}$ is Lipschitz.

Characterize the bounded linear functionals on Euclidean space $\mathbb {R}^{n}$. Show that for a sequence in $\mathbb {R}^{n}$, the concepts of strong convergence, weak convergence, and componentwise convergence are the same.

For $1 \leq p < \infty$, assume that both $\left\{ f_{n}\right\} \rightarrow h$ and $\left\{ f_{n}\right\} \rightarrow g$ weakly $L^{p}(E)$. Show that $g = h$.

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

At what point does the proof of Theorem 14 fail in the case $p = 1$ and also in the case $p = \infty$.

Theorem 14: If $1 \leq p < \infty$, then every bounded sequence in $L^{p}(E)$ has a subsequence that converges weakly to a function in $L^{p}(E)$.

Show that if a sequence in $C[a, b]$ converges weakly then it converges pointwise. Find a bounded sequence $C[a, b]$ that fails to have any weakly convergent subsequence.

Verify the assertions made regarding the convergence of the Radamacher functions.

If $\left\{ f_{n}\right\} \rightarrow f$ weakly in $L^{\infty }[a, b]$, show that

Fix real numbers $\alpha$ and $\beta$. For each natural numbers $k$ and $n$, consider the step-function $f_{n}$ defined on $I = [0,1]$ by

Show that $\left\{ f_{n}\right\}$ converges weakly in $L^{2}(I)$ to the constant function that takes the value $(\alpha +\beta ) / 2$. For $\alpha \neq \beta$, show that no subsequence of $\left\{ f_{n}\right\}$ converges strongly in $L^{2}(I)$.

Show that $\left\{ f_{n}\right\} \rightarrow f$ strongly in $L^{2}(E)$ if and only $\left\{ f_{n}\right\} \rightarrow f$ weakly in $L^{2}(E)$ and $\left\{ \left\| f_{n}\right\|_{2}\right\} \rightarrow \left\| f\right\|_{2}$.

Use Problem 20 to show that, for $1 \leq p < \infty$, a bounded sequence in $\ell^{p}$ converges weakly if and only if it converges componentwise.

Show that if a sequence of real numbers $\left\{ a_{n}\right\}$ converges to $a \in \overline{\mathbb {R}}$, then its sequence of arithmetic means also converges to $a$.

If $\Phi \in X^{*}$, prove that

For $1 < p < \infty$ define $C = \left\{ f \in L^{p}(E) \mid \left\| f\right\|_{p} \leq 1\right\}$. For $\Phi \in \left[L^{p}(E)\right]^{*}$, identify a function $f_{0} \in C$ at which the functional $\Phi : C \rightarrow \mathbb {R}$ has a minimum value.

Let $C = \left\{ f \in L^{1}[0,1] \mid \left\| f\right\|_{1} \leq 1\right\}$, and define $\Phi : C \rightarrow \mathbb {R}$ by $\Phi (f) = \int_{0}^{1} x f(x) d x$. Show that $C$ is a closed, bounded, convex subset of $L^{1}[0,1]$, that $\Phi : C \rightarrow \mathbb {R}$ is continuous and convex, but does not have a minimum value.

Let $C = \left\{ f \in C[0,1] \mid \left\| f\right\|_{\max } \leq 1\right\}$, and define $\Phi : C \rightarrow \mathbb {R}$ by $\Phi (f) = \int_{0}^{1 / 2} f(x) d x- \int_{1 / 2}^{1} f(x) d x$. Show that $C$ is a closed, bounded, convex subset of $C[0,1]$, that $\Phi : C \rightarrow \mathbb {R}$ is continuous and convex, but does not have a minimum value.

State and prove the Banach-Saks Theorem in $\ell^{2}$.

Let $m(E) < \infty$ and $p_{1}, p_{2} \in [1, \infty )$. Suppose $\varphi : \mathbb {R} \rightarrow \mathbb {R}$ is continuous and there are $c_{1}, c_{2} \geq 0$ for which $\left|\varphi (s)\right| \leq c_{1}+c_{2}\left|s\right|^{p_{1} / p_{2}}$ for all $s$. Show that

Let $C$ be a subset of a normed linear space $X$ with the property that each sequence in $C$ has a subsequence that converges strongly to a point in $C$. Show that $C$ is a closed and bounded subset of $X$.