(I'm just adding the completeness condition to $V$ from [this 2 month old question of mine](https://mathoverflow.net/questions/42731/are-coordinate-functions-on-topological-vector-spaces-always-continuous), because I realized it's relevant to whether Bill Johnson's answer to [this 4 month old question of mine](https://mathoverflow.net/questions/34967/basis-for-l-inftyr) actually answers the question.)
  
  
  
  

Let $V$ be a complete Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
  

Let $B$ be a subset of $V$ such that


$\;$ 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


and $f : B\times V \to \mathbb{K}$ be a function such that


$\;$ for all vectors $v$ in $V$, $\; \displaystyle\sum\_{b\in B} \; f(b,v)\cdot b = v$.
  
  
  

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?



Are coordinate functionals on complete vector spaces always continuous?
Consider $\ell\_1:=\ell\_1(\Bbb{N}\cup \{0\})$ 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 $e\_n^\*(x)=x(n)$ 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$^\*$ 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 $e\_n^\*$ converges weak$^\*$ to $e\_0^\*$.


