Category:Definitions/Orthonormal Bases of Vector Spaces

This category contains definitions related to Orthonormal Bases of Vector Spaces.
Related results can be found in Category:Orthonormal Bases of Vector Spaces.

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 $\struct {V, \norm {\, \cdot \,} }$ if and only if:

$(1): \quad \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is an orthogonal basis of $V$

$(2): \quad \norm {\mathbf e_1} = \norm {\mathbf e_2} = \cdots = \norm {\mathbf e_n} = 1$