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Complementary Subspace Problem


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 c_ degrees is uncomplemented in the L-infinity-space of positive integers l^infty.

Pełczyński (1960) showed that complemented subspaces of l^1, the Banach space of all absolutely summable complex sequences equipped with l_1-norm, are isomorphic to l^1.

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 C^*-algebra is necessarily linearly isomorphic to a Hilbert space.

Gowers and Maurey (1993) showed that there exists a Banach space X without nontrivial complemented subspaces.


See also

Banach Space, Complemented Subspace

This entry contributed by Mohammad Sal Moslehian

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References

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. A 121, 1-4, 1992.

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Complementary Subspace Problem

Cite this as:

Moslehian, Mohammad Sal. "Complementary Subspace Problem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ComplementarySubspaceProblem.html

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