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Let $E$ be a Banach space with basis. Is it possible to find examples such that $S \subset E$ be a complemented subspace without basis?

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  • $\begingroup$ This looks relevant. $\endgroup$ – David Mitra Feb 19 '17 at 8:02
  • $\begingroup$ The paper of Szarek, mentioned in the above link (and giving an affirmative answer to your question) is: Szarek, S. J. A Banach space without a basis which has the bounded approximation property. Acta Math. 159 (1987), no. 1-2, 81-98. $\endgroup$ – David Mitra Feb 19 '17 at 8:09
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Yes. The paper: Szarek, S. J. A Banach space without a basis which has the bounded approximation property. Acta Math. 159 (1987), no. 1-2, 81-98, provides an example heralded by the title.

A separable Banach space $X$ has the bounded approximation property if and only if $X$ is isomorphic to a complemented subspace of a space with a basis. This result can be found in Classical Banach Spaces, Vol.1, Lindenstrauss and Tzafriri (Theorem 1.e.13).

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