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topological complement
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(Definition)
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Let $X$ be a topological vector space and $M \subseteq X$ a closed subspace.
If there exists a closed subspace $N \subseteq X$ 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.
- It is known that every subspace $M \subseteq X$ has an algebraic complement, i.e. there exists a subspace $N \subseteq X$ such that $M \oplus N = X$ . The existence of topological complements, however, is not always assured.
- If $X$ is an Hilbert space, then each closed subspace $M \subseteq X$ is topologically complemented by its orthogonal complement $M^{\perp}$ , i.e.
- Moreover, for Banach spaces the converse 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.
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"topological complement" is owned by asteroid. [ full author list (2) ]
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| Also defines: |
topologically complementary, topologically complemented |
This object's parent.
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Cross-references: isomorphic, converse, Banach spaces, orthogonal complement, Hilbert space, algebraic complement, closed, subspace, topological vector space
There are 3 references to this entry.
This is version 2 of topological complement, born on 2007-09-15, modified 2009-06-27.
Object id is 9941, canonical name is TopologicalComplement.
Accessed 1626 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) | | | 46A99 (Functional analysis :: Topological linear spaces and related structures :: Miscellaneous) |
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Pending Errata and Addenda
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