Are coordinate functionals on complete vector spaces always continuous? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:19:15Z http://mathoverflow.net/feeds/question/48708 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48708/are-coordinate-functionals-on-complete-vector-spaces-always-continuous Are coordinate functionals on complete vector spaces always continuous? Ricky Demer 2010-12-09T04:32:17Z 2010-12-11T23:28:06Z <p>(I'm just adding the completeness condition to $V$ from <a href="http://mathoverflow.net/questions/42731/are-coordinate-functions-on-topological-vector-spaces-always-continuous" rel="nofollow">this 2 month old question of mine</a>, because I realized it's relevant to whether Bill Johnson's answer to <a href="http://mathoverflow.net/questions/34967/basis-for-l-inftyr" rel="nofollow">this 4 month old question of mine</a> actually answers the question.) <br><br><br><br> Let $V$ be a complete Hausdorff locally convex topological vector space over the field $\mathbb{K}$. <br> Let $B$ be a subset of $V$ such that</p> <p>$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum_{b\in B} \; c(b)\cdot b = \textbf{0}$, then $c$ is identically zero</p> <p>and $f : B\times V \to \mathbb{K}$ be a function such that</p> <p>$\;$ for all vectors $v$ in $V$, $\; \displaystyle\sum_{b\in B} \; f(b,v)\cdot b = v$. <br><br><br> Let $b$ be a member of $B$, and $g : V \to \mathbb{K}$ be given by $g(v) := f(b,v)$. Does it follow that $g$ is continuous?</p> http://mathoverflow.net/questions/48708/are-coordinate-functionals-on-complete-vector-spaces-always-continuous/49095#49095 Answer by Bill Johnson for Are coordinate functionals on complete vector spaces always continuous? Bill Johnson 2010-12-11T23:28:06Z 2010-12-11T23:28:06Z <p>Consider <code>$\ell_1:=\ell_1(\Bbb{N}\cup \{0\})$</code> as the dual of $c$, the space of convergent sequences indexed by $\Bbb{N}$, where the action is given by $e_0^*(x)=\lim_n x(n)$ and <code>$e_n^*(x)=x(n)$</code> for $n\ge 1$. Put the bw$^*$ topology on $\ell_1$ under this pairing, which is the largest locally convex topology that agrees with the weak<code>$^*$</code> topology on bounded sets. IIRC, this is a complete topology (while the weak$^*$ topology is only boundedly complete). This is an unconditional basis that is not a Schauder basis because <code>$e_n^*$</code> converges weak<code>$^*$</code> to $e_0^*$.</p>