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?
2026-04-06 11:55:53.1775476553
A complemented subspace without basis of a space with a basis
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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).