sorry if I make some grammatical faults
When I read some maths text about inner product space and norms, I see that inner product spaces induces norms. But does it mean that the inner product space induce a norm or induce THE norm ? So is a norm on an inner product space unique ? Thanks !
Each inner product determines a norm. If you know the norm and know that it comes from an inner product you can recover the inner product:
Finding an inner product given a norm
But a normed space can have different norms that come from different inner products. Consider $$ \langle (u,v) , (x,y) \rangle = ux + 2vy $$ in the plane.