Assume that we have a Banach space $X$ and a family of projections $\{P_\alpha\}_{\alpha}$ onto a finite dimensional subspaces $X_\alpha$. Let $\{e_{i,\alpha}\}_{i\in\{1,\dots,\dim X_\alpha\}}$ be a basis of $X_\alpha$.
Assume moreover that:
- $\sup_\alpha \|P_\alpha\|,\sup_\alpha \dim X_\alpha<\infty$
- $span \bigcup_\alpha X_\alpha$ is dense in $X$
- $\{e_{i,\alpha}\}_{i,\alpha}$ is an (algebraic) basis of $span \bigcup_\alpha X_\alpha$
Does it follow that:
1) $(\sum_{\alpha\in F}P_\alpha)_F$, where $F$ are finite subsets of the index set, converge to identity operator in the strong topology as $F$ grows?
2) $\forall x\in X\; (\forall\, \alpha \;P_\alpha x=0 \Rightarrow x=0)$
1) obviously implies 2). I suppose neither of these is true, but i can't think of any counterexample.