Definition
Orthonormal Basis
Let
be an inner product space. A family of vectors is an orthonormal basis if it is orthonormal and where
denotes topological closure (in finite dimensions this reduces to ). Every
then admits the expansion
Parseval Identity
For
i.e. Parseval’s identity.
Gram-Schmidt
Given a linearly independent basis, we can form linear combinations of the basis vectors to obtain an orthonormal basis.
Examples
Trivial Orthonormal Basis of
The trivial orthonormal basis of
is the family . With the usual inner product on , Every
can then be written as
Orthonormal Basis of
The standard basis of
is the family With the usual inner product,
So every vector
can be written as