📂Hilbert Space

Orthogonality, Orthogonal Sets, and Orthonormal Sets in Inner Product Spaces

Definition¹

Let $\left( X, \left\langle \cdot, \cdot \right\rangle \right)$ be an inner product space. If two elements $\mathbf{x}, \mathbf{y}\in X$ satisfy $\left\langle \mathbf{x}, \mathbf{y} \right\rangle =0$, then $\mathbf{y}$ and $\mathbf{x}$ are said to be orthogonal and denoted as follows.

$$ \mathbf{x} \perp \mathbf{y} $$

If the set of elements $X$, $\left\{ \mathbf{x}_{k} \right\}_{k\in \mathbb{N}}$, satisfies the following equation, it is called an orthogonal system or an orthogonal set.

$$ \left\langle \mathbf{x}_{k}, \mathbf{x}_{\ell} \right\rangle =0\quad \forall k\ne \ell $$

If the orthogonal system $\left\{ \mathbf{x}_{k} \right\}_{k\in \mathbb{N}}$ satisfies the following equation, it is called an orthonormal system or an orthonormal set.

$$ \left\| \mathbf{x}_{k} \right\| =1\quad \forall k\in \mathbb{N} $$

Explanation

In an inner product space, since the norm is defined as $\left\| \cdot \right\|:=\sqrt{\left\langle \cdot,\cdot \right\rangle }$, redefining the orthonormal system gives the following equation.

$$ \left\| \mathbf{x}_{k} \right\| = \left\langle \mathbf{x}_{k},\mathbf{x}_{\ell} \right\rangle = \begin{cases} 1 & \text{if}\ k=\ell \\ 0 & \text{if}\ k\ne \ell \end{cases} $$

Moreover, there is no need for the orthogonal system to be defined specifically for a countable set.

Definition²

Let $A$ be an arbitrary index set, and $\alpha$, $\beta$ be indices of $A$. If the set of elements $X$, $\left\{ \mathbf{x}_{\alpha} \right\}_{\alpha \in A}$, satisfies the following equation, it is called an orthogonal system or an orthogonal set.

$$ \left\langle \mathbf{x}_{\alpha}, \mathbf{x}_{\beta} \right\rangle =0\quad \forall \alpha \ne \beta $$

If the orthogonal system $\left\{ \mathbf{x}_{\alpha} \right\}_{\alpha \in A}$ satisfies the following equation, it is called an orthonormal system or an orthonormal set.

$$ \left\| \mathbf{x}_{\alpha} \right\| =1\quad \forall \alpha \in A $$

Explanation

Therefore, for the orthonormal system $\left\{ \mathbf{x}_{\alpha} \right\}_{\alpha \in A}$, the following equation is obtained.

$$ \left\langle \mathbf{x}_{\alpha},\mathbf{x}_{\beta} \right\rangle =\begin{cases} 1 & \text{if}\ \alpha=\beta \\ 0 & \text{if}\ \alpha \ne \beta \end{cases} $$