# Cauchy Schwarz inequality true with positive semi-definite inner product space

In one of exercises in a linear algebra book I have been asked to prove the following : "Suppose we modify the inner product definition such that $\langle u,u\rangle=0$ need not imply $u=0$." I have not Cauchy Schwarz inequality still holds in such an inner product space.

I have not understood the implication of this exercise. In the proof of the inequality we never use any such thing. Obviously, with this modified definition nothing is going to change. Am I missing something ?

• If there is a "case of equality" at the end of your statement of Cauchy-Schwarz, then that will change. – GEdgar Oct 8 '13 at 12:22
• hmm understood. – user96000 Oct 8 '13 at 14:47