Direct Sum of Subspace and Orthocomplement
Theorem
Let $H$ be a Hilbert space.
Let $M$ be a closed linear subspace of $H$.
Denote by $M^\perp$ its orthocomplement.
Then the direct sum $M \oplus M^\perp$ is isomorphic to $H$.
Proof
Assert that $U: M \oplus M^\perp \to H: \left({m, m^\perp}\right) \mapsto m + m^\perp$ is an isomorphism.
According to the definition of isomorphism, it is sufficient to check that $U$ is surjective and that:
- $\left\langle{ U \left({m, m^\perp}\right), U \left({n, n^\perp}\right) }\right\rangle_H = \left\langle{ \left({m, m^\perp}\right), \left({n, n^\perp}\right) }\right\rangle_{M \oplus M^\perp}$
First the surjectivity:
Denote by $P$ the orthogonal projection on $M$.
Then for any $h \in H$, have $h = Ph + \left({h - Ph}\right)$.
By definition of $P$, $Ph \in M$.
Furthermore, by Orthogonal Projection onto Orthocomplement, $h - Ph \in M^\perp$.
It follows that $h = U \left({Ph, h - Ph}\right)$, showing that $U$ is a surjection.
It remains to check that $U$ preserves the inner product:
\(\ds \left\langle{ U \left({m, m^\perp}\right), U \left({n, n^\perp}\right) }\right\rangle_H\) | \(=\) | \(\ds \left\langle{ m + m^\perp, n + n^\perp}\right\rangle_H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left\langle{m, n}\right\rangle_H + \left\langle{m^\perp, n}\right\rangle_H + \left\langle{m, n^\perp}\right\rangle_H + \left\langle{m^\perp, n^\perp}\right\rangle_H\) | Linearity of $\left\langle{\cdot, \cdot}\right\rangle_H$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \left\langle{m, n}\right\rangle_H + \left\langle{m^\perp, n^\perp}\right\rangle_H\) | Defining property of orthocomplement | |||||||||||
\(\ds \) | \(=\) | \(\ds \left\langle{ \left({m, m^\perp}\right), \left({n, n^\perp}\right) }\right\rangle_{M \oplus M^\perp}\) | Definition of $\left\langle{\cdot, \cdot}\right\rangle_{M \oplus M^\perp}$ |
Hence $U$ is an isomorphism, and so $M \oplus M^\perp$ is isomorphic to $H$.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Exercise $3$