Suppose h#; #i1 and h#; #i2 are inner products on v such that hv;wi1 d 0 if and only if hv;wi2 d 0. prove that there is a positive number c such that hv;wi1 d chv;wi2 for every v;w 2 v.

We take v in the vector space V different from 0. Since v is not 0, then

Lets take k>0 such that

Now, we take any vector w. We want to show that .

Since v is any non-zero vector, then this will prove that for any vectors a,b. The reason is that for any vector different from 0, lets name it x, there will exist a constant   such that for any y (this is for the same reason a constant exists for v). Since y can be anything, then it can be v. But that means that , because v also has its constant k.

Now, lets show that .  Lets take a constant c such that . We have that

Thus

Which means that

Which means that . As a consequence

Which proves what we were looking for.