how to define an inner product that inverse the original distance and have an associated norm

Question

supposing that I have an inner product $\langle a,b\rangle$ based on the normed space. I want to inverse the distance, that is if the distance between two vectors are small, under the new inner product operator it becomes bigger. It is not hard to define, say that $P(a,b) = 1 - \langle a,b\rangle$. However, since all inner product is associated with a norm, I wonder what would be a norm for this case ?

Thanks

Answer 1

That function $P$ is not an inner product, since $P(0,0)=1\neq0$.