Definition:Orthonormal Basis

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Definition



Orthonormal Basis of Vector Space

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


Also see

  • Results about orthonormal bases can be found here.