Considering a basis from two different space.

Question

For a vector space V there is a orthonormal basis. If we watch these basis from a subspace of this vector space, are they still orthonormal?

Answer 1

If you have an inner product space then the inner product structure on a subspace is induced by restricting the inner product. The restriction obviously preserves the inner product of two vectors belonging to the subspace, and hence preserves orthonormality.