The complementary subspace problem asks, in general, which closed subspaces of a Banach space are complemented
(Johnson and Lindenstrauss 2001).
Phillips (1940) proved that the Banach space of all complex sequences converging to zero together with the supremum
norm
is uncomplemented in the L-infinity-space of
positive integers .
Pełczyński (1960) showed that complemented subspaces of , the Banach space of all
absolutely summable complex sequences equipped with -norm, are isomorphic to .
In 1971, Lindenstrauss and Tzafriri (1977) proved that every infinite-dimensional Banach space that is not isomorphic to a Hilbert
space contains a closed uncomplemented subspace.
Pisier (1992) established that any complemented reflexive subspace of a -algebra is necessarily linearly isomorphic to a Hilbert
space.
Gowers, W. T. and Maurey, B. "The Unconditional Basic Sequence Problem." J. Amer. Math. Soc.6, 851-874, 1993.Johnson,
W. B. and Lindenstrauss, J. (Eds.). Handbook
of the Geometry of Banach Spaces, Vol. 1. Amsterdam, Netherlands: North-Holland,
2001.Lindenstrauss, J. and Tzafriri, L. Classical
Banach Spaces. I. Sequence Spaces. New York: Springer-Verlag, 1977.Pełczyński,
A. "Projections in Certain Banach Spaces." Studia Math.19,
209-228, 1960.Phillips, R. S. "On Linear Transformations."
Trans. Amer. Math. Soc.48, 516-541, 1940.Pisier, G. "Remarks
on Complemented Subspaces of Von Neumann Algebras." Proc. Roy. Soc. Edinburgh
Sect. A121, 1-4, 1992.
is uncomplemented in the L-infinity-space of
positive integers 
.
, the Banach space of all
absolutely summable complex sequences equipped with 
-norm, are isomorphic to 
.
-algebra is necessarily linearly isomorphic to a Hilbert
space.
without nontrivial complemented subspaces.
