projections and closed subspaces
Theorem 1  Let $X$ be a Banach space^{} and $M$ a closed subspace. Then,

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$M$ is topologically complemented in $X$ if and only if there exists a continuous^{} projection onto $M$.

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Given a topological complement $N$ of $M$, there exists a unique continuous projection $P$ onto $M$ such that $P(x+y)=x$ for all $x\in M$ and $y\in N$.
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The projection $P$ in the second part of the above theorem is sometimes called the projection onto $M$ along $N$.
The above result can be further improved for Hilbert spaces^{}.
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Theorem 2  Let $X$ be a Hilbert space and $M$ a closed subspace. Then, $M$ is topologically complemented in $X$ if and only if there exists an orthogonal projection onto $M$ (which is unique).
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Since, by the orthogonal decomposition theorem, a closed subspace of a Hilbert space is always topologically complemented by its orthogonal complement^{} ($X=M\oplus {M}^{\u27c2}$), it follows that
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Corollary  Let $X$ be a Hilbert space and $M$ a closed subspace. Then, there exists a unique orthogonal projection onto $M$. This establishes a bijective^{} correspondence between orthogonal projections and closed subspaces.
Title  projections and closed subspaces 

Canonical name  ProjectionsAndClosedSubspaces 
Date of creation  20130322 17:52:57 
Last modified on  20130322 17:52:57 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  5 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 46C07 
Classification  msc 46B20 
Synonym  projection along a closed subspace 
Synonym  orthogonal projections onto Hilbert subspaces 