Definition:Orthonormal Basis of Vector Space

Definition
Let $\struct {V, \norm {\, \cdot \,} }$ be a normed vector space.

Let $\BB = \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be a basis of $\struct {V, \norm {\, \cdot \,} }$.

Then $\BB$ is an orthonormal basis of $\struct {V, \norm {\, \cdot \,} }$ :
 * $(1): \quad \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is an orthogonal basis of $V$
 * $(2): \quad \norm {\mathbf e_1} = \norm {\mathbf e_2} = \cdots = \norm {\mathbf e_n} = 1$

Also known as
An orthonormal basis is also known as a Cartesian basis, particularly when it is used as the basis of a Cartesian coordinate system.

Also see

 * Definition:Cartesian Coordinate System