topological complement

0.0.1 Definition

Let X be a topological vector spaceMathworldPlanetmath and MX a closed ( subspacePlanetmathPlanetmath.

If there exists a closed subspace NX such that


we say that M is topologically complemented.

In this case N is said to be a topological complement of M, and also M and N are said to be topologically complementary subspaces.

0.0.2 Remarks

  • It is known that every subspace MX has an algebraic complement, i.e. there exists a subspace NX such that MN=X. The existence of topological complements, however, is not always assured.

  • If X is an Hilbert spaceMathworldPlanetmath, then each closed subspace MX is topologically complemented by its orthogonal complementMathworldPlanetmath M, i.e.

  • Moreover, for Banach spacesMathworldPlanetmath the converseMathworldPlanetmath of the last paragraph also holds, i.e. if each closed subspace is topologically complemented then X is isomorphic a Hilbert space. This is the Lindenstrauss-Tzafriri theorem (

Title topological complement
Canonical name TopologicalComplement
Date of creation 2013-03-22 17:32:31
Last modified on 2013-03-22 17:32:31
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 5
Author asteroid (17536)
Entry type Definition
Classification msc 46A99
Classification msc 15A03
Defines topologically complementary
Defines topologically complemented