Orthonormal Basis

linear-algebra

Definition

Orthonormal Basis

Let $(V,⟨\cdot |\cdot ⟩)$ be an inner product space. A set of vectors $\{e_i{\}}_{i\in I}\subseteq V$ is an orthonormal basis if:

1. Orthogonality: $⟨e_i|e_j⟩=0$ for all $i\ne j$
2. Normalisation: $⟨e_i|e_j⟩=1$ for all $i=j$
3. Completeness: $\overline{\text{span}\{e_i\}}=V$

Equivalently:

$$⟨e_i|e_j⟩={\delta }_{ij}$$

where ${\delta }_{ij}$ is the Kronecker delta.

Every $v\in V$ admits the expansion:

$$v=\sum _i⟨e_i|v⟩e_i$$

with coefficients $c_i=⟨e_i|v⟩$ satisfying $\Vert v{\Vert }^2={\sum }_i|c_i{|}^2$ (Parseval’s identity).