Exercises 1.1 Exercises 1.1 Prove that . One has . Taking the square root of both sides gives the result.
Equality holds precisely when one is a nonnegative multiple of the other.
This is a consequence of the analogous assertion of the next problem. Prove that . The first assertion is the triangle inequality. I claim that equality holds precisely when one vector is a non-positive multiple of the other. If for some real , then substituting shows that the inequality is equivalent to and clearly equality holds if a is non-positive.