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 ?
No, you probably aren't missing anything. The implication of the exercise was making you read the given proof of Cauchy-Schwarz again fully and checking that the definiteness of the inner product was never used.