If then for all i. Therefore . Thus .
For the reverse inequality, define y by for each i (in other words +1 if is positive, –1 if it is negative, and 0 if it is 0). Then and .
If the scalars are complex then the argument is similar, with complex conjugates thrown in where necessary. For the second part, you have to define to be the complex number of absolute value 1 such that is real and positive.