projections and closed subspaces

Theorem 1 - Let $X$ be a Banach space and $M$ a closed subspace. Then,

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

• 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 ($X=M\oplus M^{⟂}$), 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.