Orthonormal Basis

Let be an Inner Product Space.
A sequence is an orthonormal basis if:

1. it is an Orthonormal Sequence
2. The span is dense in

We use the same terminology for a finite sequence
Gram-Schmidt process applies

Corollary

Let be a Separable Inner Product Space.
Then has an orthonormal basis.

Corollary

Let be an -dimensional inner product space.
Then has a Isometric Isomorphism to .