1.PROJECT.Norms. (a)Vector norms are equivalent, that is, they are related by double inequalities; for instance, (1) || xllco 5 ||xl|l, S n | X ||.o (2) | xllsl|xllo 5 || x ||| n 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 (1). (b)The Cauchy-Schwarz inequality is |x" y| s || x ||2|| y || : It is very important. Use it to prove (3) || x|l2 5||x|l, s vn || x|l2 (4) to livell , 5 || xllz 5 || xe Il (c) Formula (6) is often more practical than (5). Derive (6) from (5). ||AX|| ||A|| = max (x0) ||A| (5) |A||: = max ||AX|| (6) |x)=1