# Complemented subspaces constructed from finite pieces- part II

This is a follow up to: Complemented subspace constructed from finite pieces

Suppose $$Y=\overline{\cup E_n}$$ is a closed subspace of a separable Banach space X, 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 $$X$$?

In light of the answer to the previous question, a related question would be the following:

Is $$c_0$$ complemented in every separable subspace of $$l_\infty$$ that contains it. I suspect the answer is no, but cannot think of a counterexample.

Added on January 5, 2018. Many other examples giving a negative answer to the first question can be constructed combining the result of Zippin [Israel J. Math. 26 (1977), no. 3-4, 372–387] stating that any separable infinite-dimensional Banach space which is not isomorphic to $$c_0$$ can be embedded into a separable Banach space as an uncomplemented subspace, and the well-known fact that a Banach space can represented as a closure of the union of increasing sequence of finite-dimensional subspaces which are uniformly close to $$\ell_\infty^n$$ without being isomorphic to $$c_0$$ (see, e.g., very exotic examples in Bourgain, Pisier [Bol. Soc. Brasil. Mat. 14 (1983), no. 2, 109–123].)
• In the case that $X$ is reflexive (but perhaps non-separable), how can we make sure that a weak limit point $P$ of the projections $P_n:X\to E_n$ is a projection onto $Y$? The range of $P$ might contain $Y$ properly. Am I missing something simple? Nov 29, 2021 at 12:59