# Complemented subspace constructed from finite pieces

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 $$X$$?

• Why the Hilbert spaces tag? The question is trivially yes for Hilbert spaces Dec 25 '18 at 0:51
• Removed Hilbert space tag, indeed irrelevant. Dec 25 '18 at 0:54

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".