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Comput. Phys.} % \begin{document} \title{Practice Problem set 3} \maketitle \begin{enumerate} \item Suppose that $A$ is closed convex compact subset of $X$ a complete vector space. Suppose that $F=\{ x_{0} \}$ is an extreme set of $A$. Show that $x_{0} \in \mathcal{E}(A)$. \item Prove the inverse mapping theorem. If $T:X\to Y$ is linear, bounded and a bijection. Then show that $T^{-1}$ is bounded. \item Suppose that $T:X\to Y$ is linear and bounded. Show that $T$ is an open map, i.e. $T(A)$ is open for every open set $A$, if and only if $T(B_{1}(0))$ contains a ball centered at the origin. \item Suppose that $B$ is a closed extreme subset of $A$, a compact closed convex set. Suppose that $\ell \in X^{\ast}$ is a bounded linear functional. Show that $$ F = \{ x\in B \, \, | \, \, \ell(x) = \sup_{y\in B} \ell(y) \} \, , $$ is a non-empty, closed, convex extreme subset of $A$. \item Suppose that $X = L^{1}[0,1]$. Show that $\overline{B_{1}(0)}\subset X$ has no extreme points. \item Show that $\ell^{p}(\bN)$, $0