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Orthogonality, Orthogonal Sets, and Orthonormal Sets in Inner Product Spaces 📂Hilbert Space

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

Definition1

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

x⊥y \mathbf{x} \perp \mathbf{y}

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

⟨xk,xℓ⟩=0∀k≠ℓ \left\langle \mathbf{x}_{k}, \mathbf{x}_{\ell} \right\rangle =0\quad \forall k\ne \ell

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

∥xk∥=1∀k∈N \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.

∥xk∥=⟨xk,xℓ⟩={1if k=ℓ0if k≠ℓ \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 AA be an arbitrary index set, and α\alpha, β\beta be indices of AA. If the set of elements XX, {xα}α∈A\left\{ \mathbf{x}_{\alpha} \right\}_{\alpha \in A}, satisfies the following equation, it is called an orthogonal system or an orthogonal set.

⟨xα,xβ⟩=0∀α≠β \left\langle \mathbf{x}_{\alpha}, \mathbf{x}_{\beta} \right\rangle =0\quad \forall \alpha \ne \beta

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

∥xα∥=1∀α∈A \left\| \mathbf{x}_{\alpha} \right\| =1\quad \forall \alpha \in A

Explanation

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

⟨xα,xβ⟩={1if α=β0if α≠β \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}

  1. Ole Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (2010), p66-67 ↩︎

  2. Walter Rudin, Real and Complex Analysis (3rd Edition, 1987), p82-83 ↩︎