I'm having trouble with the following problem:

Let be a real vector space equipped with an inner product (and thus with a norm induced by that inner product).

Let be a bijective linear transformation such that if , then .

Given an orthonormal basis is , it clearly is the case that is an orthogonal basis of and hence is an orthonormal basis.

I now have to show that , but everything I've tried so far has lead me to a dead end.

Any hint or tip from someone who knows how to prove this would be greatly appreciated (I'm not looking for someone to post a complete solution).