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Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm. and let $\ell_0(x) = \lim_{n \rightarrow \infty} x_n$, for convergent sequences. Show that the sett L consisting of all continuous extensions of $\ell_0$ to $\ell^\infty$ is closed in the weak* topology on $(\ell^\infty)'$.

The extension is done by Hahn-Banach, but how do I show that something is closed in the weak*? Can I use sequentially closed here? are the topology metriceble? I can show that this is not a Hilbert space. Is it reflexive? Im a little bit unsure about all this weak* stuff, Please help me out and merry christmas!

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To show $L$ is weak-* closed, you want to show that its complement is weak-* open, i.e. any $\phi \in (\ell^\infty)' \backslash L$ has a weak-* neighborhood disjoint from $L$. In fact, if $\phi \in (\ell^\infty)' \backslash L$ there is some sequence $s$ that converges to a limit $m$ that is not $\phi(s)$. Consider $\{\psi \in (\ell^\infty)': |\psi(s) - \phi(s)|<\epsilon\}$.

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    @RobertIsrael, THX2013-01-02