Proof
We prove the statement by induction over , that is, we construct successively a family of orthonormal vectors spanning the same linear subspaces. For
,
we just have to normalize , that is, we replace it by
.
Now suppose that the statement is already proven for . Let a family of orthonormal vectors fulfilling
be already constructed. We set
-
Due to
this vector is orthgonal to all , and also
-
holds. By normalizing , we obtain .