(a) Vector norms in our text are equivalent, that is, they are related by double inequalities; for instance,

Hence if for some x, one norm is large (or small), the other norm must also be large (or small). Thus in many investigations the particular choice of a norm is not essential. Prove (18).

(b) The Cauchy-Schwarz inequality is

It is very important. Use it to prove.

(c) Formula (10) is often more practical than (9). Derive (10) from (9).

(d) Matrix norms. Illustrate (11) with examples. Give examples of (12) with equality as well as with strict inequality. Prove that the matrix norms (10), (11) satisfy the axioms of a norm 

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(a) ||x S || | s || |. |(18) (b) |xTyl = ||x||2 |y |2.