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(2006 / vol.9 / no.1)
A survey of the complemented subspace problem
moslehian
Pages. 91-98     



The complemented subspace problem asks, in general, which closed
subspaces $M$ of a Banach space $X$ are complemented; i.e. there
exists a closed subspace $N$ of $X$ such that $X=Moplus N$? This
problem is in the heart of the theory of Banach spaces and plays
a key role in the development of the Banach space theory. Our aim
is to investigate some new results on complemented subspaces, to
present a history of the subject, and to introduce some open
problems.



1. Introduction 2. Complementary subspace problem and related results 3.Schroeder-Bernstein Problem 4.Basis and complemented subspaces 5. Approximation property and complemented subspaces 5. Complemented minimal subspaces 6. Quasi-complemented subspaces 7. Weakly complemented subspaces 8. contractively complemented subspaces 9. Prime Banach spaces and complemented subspaces 10. Complemented subspaces of topological products and sums 11. Some interesting problems



Complemented subspace, Schauder basis, basis, $L_1$-predual space, weakly complemented subspace, quasi-complemented subspace, complementary minimal subspace, prime space, sequence spaces.



Primary 46B20; secondary 46B25