topological complement
0.0.1 Definition
Let be a topological vector space and a closed (http://planetmath.org/ClosedSet) subspace.
In this case is said to be a topological complement of , and also and are said to be topologically complementary subspaces.
0.0.2 Remarks
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It is known that every subspace has an algebraic complement, i.e. there exists a subspace such that . The existence of topological complements, however, is not always assured.
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If is an Hilbert space, then each closed subspace is topologically complemented by its orthogonal complement , i.e.
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Moreover, for Banach spaces the converse of the last paragraph also holds, i.e. if each closed subspace is topologically complemented then is isomorphic a Hilbert space. This is the Lindenstrauss-Tzafriri theorem (http://planetmath.org/CharacterizationOfAHilbertSpace).
Title | topological complement |
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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 |