Complemented Subspace

Let X be a normed space, M and N be algebraically complemented subspaces of X (i.e., M+N=X and M intersection N={0}), pi:X->X/M be the quotient map, phi:M×N->X be the natural isomorphism (x,y)|->x+y, and P:X->M,P(x+y)=x,x in M,y in N be the projection of X on M along N. Then the following statements are equivalent:

1. phi is a homeomorphism.

2. M and N are closed in X and pi|_N is a homeomorphism.

3. M and N are closed and P:X->M is a bounded projection.

The subspaces M and N are called topologically complemented or simply complemented if each of the above equivalent statements holds (Constantinescu 2001, Meise and Vogt 1997).

Every finite dimensional subspace is complemented and every algebraic complement of a finite codimension subspace is topologically complemented. In a Banach space X, two closed subspace are algebraically complemented if and only if they are complemented.

There are uncomplemented closed subspaces. For example, let X be the disk algebra, i.e., the space of all analytic functions on {z in C:|z|<1} which are continuous on the closure of D. Then the subspace of C(T) consisting of the restrictions of functions of X to T={z in C:|z|=1} is not complemented in X (Hoffman 1988).

The problems related to complemented subspaces are in the heart of the theory of Banach spaces and are more than fifty years old (Johnson and Lindenstrauss 2001).

See also

This entry contributed by Mohammad Sal Moslehian

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

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