We show that if X is an infinite-dimensional Banach space in which every finite-dimensional subspace is λ-complemented with λ ≤ 2 then X is (1 + C√(λ-1))-isomorphic to a Hilbert space, where C is an absolute constant; this estimate (up to the constant C) is best possible. This answers a question of Kadets and Mityagin from 1973. We also investigate the finite-dimensional versions of the theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-3-2,
author = {N. J. Kalton},
title = {The complemented subspace problem revisited},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {223-257},
zbl = {1162.46013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-3-2}
}
N. J. Kalton. The complemented subspace problem revisited. Studia Mathematica, Tome 187 (2008) pp. 223-257. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-3-2/