Complemented subspace

Summary

A closed subspace of a Banach space is complemented when it has a closed complementary subspace. Equivalently, it is complemented when it is the range of a bounded linear projection. In linear algebra every subspace has a complement, but in Banach spaces a closed complement may fail to exist. Complemented subspaces are the pieces into which a Banach space splits as a direct sum. Much of the isomorphic theory of Banach spaces studies them.