Orthogonal Projection in Hilbert Spaces 📂Hilbert Space

Orthogonal Projection in Hilbert Spaces

Definition¹

Let’s assume a closed subspace $V$ of a Hilbert space $H$ is given.

When $\mathbf{v} \in H$ is represented as $\mathbf{v} = \mathbf{v}_{1} + \mathbf{v}_{2}$ with respect to $\mathbf{v}_{1} \in V$ and $\mathbf{v}_{2} \in V^{\perp}$, a surjection $P :H \to V$ that satisfies the following is called an orthogonal projection.

$$ P \mathbf{v} = \mathbf{v}_{1} $$

Explanation

Orthogonal projection has the following properties:

$P$ is linear, bounded, and $| P | = 1$.
$P$ is a self-adjoint operator, i.e., $P^{ \ast } = P$.
$P$ is idempotent, which means $P^{2} = P$.

Orthogonal projection extended to the Hilbert space naturally covers orthogonal projection in matrix algebra, and it feels a bit more abstract in its definition.

Theorem

Let’s assume a [normalized orthogonal basis] $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}}$ of the closed subspace $V$ of the Hilbert space $H$ is given. For every $\mathbf{v} \in H$, orthogonal projection $P : H \to V$ can be represented as follows.

$$ P \mathbf{v} = \sum_{k=1}^{\infty} \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle \mathbf{e}_{k} $$

Proof

Since $\left\{ \mathbf{e}_{k} \right\}_{k=1}^{\infty}$ is a basis of $V$, $P \mathbf{v} \in V$ is represented as follows for $\left\{ a_{k} \right\}_{k=1}^{\infty} \subset \mathbb{C}$ not equal to $a_{1} = \cdots = 0$.

$$ P \mathbf{v} = \sum_{k =1}^{\infty} a_{k} \mathbf{e}_{k} $$

Because of the orthogonality of $\left\{ \mathbf{e}_{k} \right\}_{k=1}^{\infty}$, $\left\langle \mathbf{e}_{i} , \mathbf{e}_{i} \right\rangle = 1$ and therefore $\left\langle \mathbf{e}_{i} , \mathbf{e}_{j} \right\rangle = 0$ for $i \ne j$.

$$ \left\langle P \mathbf{v} ,P \mathbf{v} \right\rangle = \sum_{k =1}^{\infty} a_{k}^{2} $$

Meanwhile, due to the property $P^{ \ast }=P$ and $P^{2} = P$ from $P \mathbf{v} = \sum_{k =1}^{\infty} a_{k} \mathbf{e}_{k}$,

$$ \left\langle P \mathbf{v} ,P \mathbf{v} \right\rangle = \left\langle \mathbf{v} ,P^{ \ast }P \mathbf{v} \right\rangle = \left\langle \mathbf{v} ,P \mathbf{v} \right\rangle = \sum_{k =1}^{\infty} a_{k} \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle $$

Hence

$$ \sum_{k =1}^{\infty} a_{k}^{2} = \sum_{k =1}^{\infty} a_{k} \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle $$

In summary,

$$ \sum_{k =1}^{\infty} a_{k} \left( a_{k} - \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle \right) = 0 $$

Therefore, for every $k \in \mathbb{N}$, $a_{k} = \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle$ must hold.

$$ P \mathbf{v} = \sum_{k=1}^{\infty} \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle \mathbf{e}_{k} \qquad , \mathbf{v} \in H $$

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