📂Hilbert Space

Gram-Schmidt Orthogonalization in Separable Hilbert Spaces

Theorem¹

Every separable Hilbert space has an orthonormal basis.

Proof

Strategy: Essentially the same as Gram-Schmidt orthonormalization in finite-dimensional vector spaces. Since the existence of a basis is not guaranteed in general Hilbert spaces unlike finite-dimensional vector spaces, it is necessary to first choose an orthogonal basis $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$ due to separability, before proceeding with orthonormalization.

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$$ \overline{\text{span}} \left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} = H $$

If Hilbert space $H$ is a separable space, then there exists $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} \subset H$ satisfying the above. Let’s define $\left\{ \mathbf{e}_{k} \right\}_{k=1}^{n}$ as follows.

$$ \begin{align*} \mathbf{e}_{1} :=& {{ \mathbf{v}_{1} } \over { \left\| \mathbf{v}_{1} \right\| }} \\ \mathbf{e}_{2} :=& {{ \mathbf{v}_{2} - \left\langle \mathbf{v}_{2} , \mathbf{e}_{1} \right\rangle \mathbf{e}_{1} } \over { \left\| \mathbf{v}_{2} - \left\langle \mathbf{v}_{2} , \mathbf{e}_{1} \right\rangle \mathbf{e}_{1} \right\| }} \\ \mathbf{e}_{n+1} :=& {{ \mathbf{v}_{n+1} - \sum_{k=1}^{n} \left\langle \mathbf{v}_{n+1} , \mathbf{e}_{k} \right\rangle \mathbf{e}_{k} } \over { \left\| \mathbf{v}_{n+1} - \sum_{k=1}^{n} \left\langle \mathbf{v}_{n+1} , \mathbf{e}_{k} \right\rangle \mathbf{e}_{k} \right\| }} \end{align*} $$

Since $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$ is an orthogonal basis of $H$, we have $\left\langle \mathbf{e}_{i} , \mathbf{e}_{j} \right\rangle = \delta_{ij}$, and for all $n \in \mathbb{N}$, the following holds.

$$ \text{span} \left\{ \mathbf{v}_{k} \right\}_{k = 1}^{n} = \text{span} \left\{ \mathbf{e}_{k} \right\}_{k =1}^{n} $$

Therefore

$$ \overline{\text{span}} \left\{ \mathbf{e}_{k} \right\}_{k = 1}^{\infty} = \overline{\text{span}} \left\{ \mathbf{v}_{k} \right\}_{k =1}^{\infty} = H $$

Equivalence conditions for an orthonormal basis: Suppose $H$ is a Hilbert space. For the orthonormal system $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}} \subset H$ of $H$, the following are all equivalent.

• (i): $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}} \subset H$ is an orthonormal basis of $H$.
• (v): $\overline{\text{span}} \left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}} = H$

Since $\overline{\text{span}} \left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}} = H$, $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}}$ is a basis of $H$, specifically, since $\left\langle \mathbf{e}_{i} , \mathbf{e}_{j} \right\rangle = \delta_{ij}$, it is an orthonormal basis.

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