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\begin{document}
\title{Assignment 6: Dual spaces}

\date{Assigned Wed 05/09. Due Wed 05/16.}
\maketitle

\begin{questions}

\item Show that Lemma 4 from the handout (solution of 3(b) from the midterm) is false if $\alpha = \beta$.

\item Let $X$ be the completion of periodic $C^1$ functions with inner product defined by $\ip{f}{g} = \int f\bar{g} + \int f' \bar g'$.
\begin{parts}
\item Show that there exists $D \in B(X, \Lpspace{2})$ such that for all $C^1$ functions $f \in X$, $Df = f'$. Compute the adjoint of the operator $i D$.
\item Show that there does not exist $D \in B(\Lpspace{2}, \Lpspace{2})$ such that for all $C^1$ functions $D f = f'$.
\item (Unrelated) Show that $X$ is isomorphic to $H^1$.
\end{parts}

\item
\begin{parts}
\item Let $K:[0,1] \times [0,1] \to \C$ be continuous. If $f \in \Lpspace{p}([0,1])$, then define $Tf$ by $Tf(x) = \int K(x,y) f(y) \,dy$. Show that $T \in B(\Lpspace{p}, \Lpspace{p})$. When $p = 2$, compute $T^*$.
\item Let $f \in \Lpspace{1}([0,1])$. Define $T:\Lpspace{p} \to \Lpspace{p}$ by $T(g) = f * g$. When $p = 2$, compute $T^*$.
\end{parts}

\item
\begin{parts}
\item Show that $(\lpspace{1})^*$ is (isometrically) isomorphic to $\lpspace{\infty}$.
\item Let $c_0 \subset \lpspace{\infty} = \{ (x_n)_n \st (x_n)_n \to 0 \}$. Show that $c_0^* \approx \lpspace{1}$.
\item Suppose there exists $\Lambda \in (\lpspace{\infty})^*$ such that $\Lambda((x_n)_n) = \lim x_n$ whenever the limit exists. Show that there does not exist $(a_n) \in \lpspace{1}$ such that $\Lambda(x) = \sum a_n x_n$ for all $x = (x_n)_n \in \lpspace{\infty}$. \noteb{Some junk set theory will show the existence of such $\Lambda$. Hence this problem shows that $(\lpspace{\infty})^* \supsetneq \lpspace{1}$.}
\item Let $1 < p < \infty$, and $q = \frac{p}{p-1}$. Let $\phi_p: \lpspace{q} \to (\lpspace{p})^*$ be the isomorphism defined by $\phi(a)(x) = \sum a_i x_i$, where $a = (a_n) \in \lpspace{q}$, and $x = (x_n) \in \lpspace{p}$. For any Banach space $X$, let $i: X \to X^{**}$ be the canonical embedding defined by $i(x)(x^*) = x^*(x)$ for all $x^* \in X^*$. Show that $\phi_q\inv \circ \phi^*_p$ is the inverse of the canonical embedding $i:\lpspace{p} \to (\lpspace{p})^{**}$. \noteb{Note that the arguments from class only say that $(\lpspace{p})^{**}$ is isomorphic to $\lpspace{p}$. There exist pathological examples of Banach spaces such that $X^{**}$ is isomorphic to $X$, but the canonical embedding is not an isomorphism. When the canonical embedding is an isomorphism (which is generally a desirable thing), the Banach space is called \textit{reflexive}.} 
\end{parts}

\item Define $\varphi: \Lpspace{q} \to (\Lpspace{p})^*$ by $\varphi(f)(g) = \int fg$. Show that $\varphi$ is an isometric embedding. \noteb{$\varphi$ is actually an isomorphism, but I don't know a proof of this without using the Radon-Nikodym theorem.}

\item Using the axiom of choice one can show that for any Banach space $X$,
$\norm{x}_X = \sup\limits_{\norm{x^*}_{X^*} = 1} \abs{x^*(x)}$. (Note the `duality' of this statement with the definition of the norm in $X^*$: $\norm{x^*}_{X^*} = \sup\limits_{\norm{x}_X = 1} \abs{x^*(x)}$). Since I can't in good conscience prove this in class, you prove this explicitly for the following Banach spaces:
\begin{mcparts}
\item $\lpspace{p}$, when $1 \leq p \leq \infty$.
\item $\Lpspace{p}$, when $1 \leq p \leq \infty$.
\end{mcparts}

\item \textit{(Youngs inequality)} Suppose $\frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r}$. If $f, g \in C^0$, show that $\Lpnorm{f * g}{r} \leq \Lpnorm{f}{p} \Lpnorm{g}{q}$. \hintb{This is quite hard to prove directly. The trick here is to pick any $h \in C^0([0,1])$, and show that $\abs{\int (f*g)(x) h(x) \,dx} \leq \Lpnorm{f}{p} \Lpnorm{g}{q} \Lpnorm{h}{r'}$, where $r' = \frac{r}{r-1}$, and then use the previous problem.}
\end{questions}
\end{document}