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 $X$?
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$\begingroup$ Why the Hilbert spaces tag? The question is trivially yes for Hilbert spaces $\endgroup$– Yemon ChoiCommented Dec 25, 2018 at 0:51
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$\begingroup$ Removed Hilbert space tag, indeed irrelevant. $\endgroup$– user129564Commented Dec 25, 2018 at 0:54
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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".
answered Dec 25, 2018 at 0:54
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$\begingroup$ Thank you. Do you also have a separable example? $\endgroup$ Commented Dec 25, 2018 at 0:55
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$\begingroup$ I can't think of one off the top of my head $\endgroup$ Commented Dec 25, 2018 at 1:03
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$\begingroup$ Thank you, maybe I will post a separate question. $\endgroup$ Commented Dec 25, 2018 at 1:04