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\begin{document}
\title{Problem set 3}
\author{Due date: Mar 5}
\maketitle

\begin{enumerate}
\item Suppose that $\ell^{1}(\bN)$ is the space of sequences which are absolutely summable, i.e.
$$
\{ a_{n} \} \in \ell^{1}(\bN) \quad \text{if} \quad \sum_{n=1}^{\infty} |a_{n} | < \infty \, . 
$$ 
Suppose that $c_{0}(\bN)$ is the space of sequences that converge to $0$, i.e.
$$
\{ a_{n} \} \in c_{0}(\bN) \quad \text{if} \quad \lim_{n\to\infty} a_{n} = 0 \, .
$$
Show that $c_{0}(\bN)^{\ast} = \ell^{1} (\bN)$.
\item Suppose that $\ell^{\infty}(\bN)$ is the space of sequences which are bounded, i.e.
$$
\{ a_{n} \} \in \ell^{\infty}(\bN) \quad \text{if} \quad \sup_{n} |a_{n}| < \infty \, .
$$
Show that $\ell^{\infty}(\bN)$ is not separable.
\item Suppose that $c\subset \ell^{\infty}(\bN)$ is the space of sequences that converge, i.e.
$$
\{ a_{n} \} \in c \quad \text{if} \quad \lim_{n\to\infty} a_{n} \quad \text{exists} \, .
$$  
Show that $c$ is a closed subspace.
\item (Optional) Recall that the dual of $C[0,1]$ is the space of Borel measures on the interval $[0,1]$. 
Construct a bounded linear functional in $\left(C[0,1] \right)^{\ast}$ which does not attain its norm.
\item Let
$$
\ell(f) = f(x_{0}) \, ,
$$
denote a linear functional in $(C[0,1])^{\ast}$ where $0<x_{0}<1$. Show that $\ell$ is a bounded and
find the norm of $\ell$.
\item Suppose that $X=L^{2}[-1,1]$. For each scalar $\alpha$, let 
$$
E_{\alpha} := \{ f\in C[-1,1], f(0) = \alpha \} \, .
$$
Show that
\begin{enumerate}
\item Each $E_{\alpha}$ is convex and dense in $X$
\item For $\alpha \neq \beta$, $E_{\alpha}$, $E_{\beta}$ are disjoint but there is no continuous functional on $\ell$ 
on $X$ such that
$$
\sup_{f \in E_{\alpha}} \ell(f) \leq \inf_{f\in E_{\beta}} \ell(f) \, .
$$
\end{enumerate}
Explain why geometric Hahn-Banach could not be employed
\item Suppose that $\cP$ is the space of all polynomials in one variable with real coefficients. Let the subset $A$ consist
of polynomials with negative leading coefficients, and let the subset $B$ consist of polynomails with all non-negative coefficients. Show that $A$ and $B$ are disjoint convex subsets of $\cP$. 
Further, show that there does not exist a nonzero linear functional $\ell$ on $\cP$ such that
$$
\ell(a) \leq \ell(b) \quad \forall a\in A \, ,b\in B \, .
$$
(Hint: assume that for some $C\in \R$, one has $\ell(a)\leq C\leq \ell(b)$, $a\in A$, $b\in B$;note that $0\in B$ and that $C\leq 0$
and consider monomials to show that $C\geq 0$.
\item Construct two closed disjoint convex sets $K_{1}$ and $K_{2}$ in $\R^{2}$ that cannot be strictly separated, i..e
there does not exist a bounded linear functional $\ell$ such that
$$
\sup_{x\in K_{1}} \ell(x) < \inf_{y\in K_{2}} \ell(y) \, .
$$ 
\item \begin{enumerate}
\item Suppose that $T:X\to Y$ and $S:Y\to Z$ are bounded, where $X,Y,Z$ are Banach spaces. Show that $(ST)^{\ast} = T^{\ast} S^{\ast}$
\item Suppose that $S,T:X\to Y$ are bounded, where $X,Y$ are Banach spaces and suppose that $a,b \in \R$. Show that $(aS+bT)^{\ast} = aS^{\ast} + bT^{\ast}$.
\item Suppose that $T^{-1}:Y\to X$ exists and is bounded. Show that $\left(T^{-1}\right)^{\ast} = \left(T^{\ast}\right)^{-1}$.
\end{enumerate}
\end{enumerate}
\end{document}