Suppose $Y=\overline{\cup E\_n}$ is a closed subspace of a Banach space, where each $E\_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E\_n\subseteq E\_{n+1}$. Can one conclude that $Y$ is complemented in $X$? 



Complemented subspace constructed from finite pieces
No. Take $X=\ell\_\infty({\bf N})$ and take $E\_n = \operatorname{span}(e\_1,\dots, e\_n)$. Then $Y=c\_0({\bf N})$ which is well-known – by a non-trivial argument – to be uncomplemented in $X$ (in the sense of Banach spaces). Look up "Phillips's Lemma".


