Orthonormal sets and inner products

Question

How do I prove that a given basis $B$ is an orthonormal set relative to a given inner product $\langle x,y\rangle$?

I am unsure if I should use the Gram-Schmidt process.

Answer 1

Gram-Schmidt is for taking a set of vectors (usually a basis) and orthonormalizing them. If you want to prove that a given basis is orthonormal, you just need to show that the pairwise inner products of the elements of the basis are zero, and that they each magnitude 1.

Answer 2

To verify that a given basis $B$ is orthonormal, you need to first verify that, for any distinct $x,y\in B$, we have $\langle x,y\rangle=0$, so $B$ is orthogonal. Next you need to verify that, for any $x\in B$, we have $\langle x,x\rangle=1$, so each vector in $B$ is a unit vector.

The Gram-Schmidt process is to convert a given basis into an orthogonal (or orthonormal if you want to) basis which generates the same space.