If we start with an inner product on a vector space, there is a unique induced norm on it, defined by
$$|| x ||=\sqrt{\langle x,x\rangle}$$
With this definition, many inequalities and theorems can be derived.
My question is: If we choose another definition of norm, would the norm and the inner product be consistent? My guess is that the vector space would still make sense, but we will lose a lot of nice properties since there is no relationship between thr inner product and the norm.