Problem Sheet 2 Exercise 1:PLet ⌦ := [0, 1]. For f : ⌦ ! K and a partition1 Z = {x0 , x1 , ..., xn }, n 2 N, of ⌦ define VZ (f ) := ni=1 |f (xi ) f (xi 1 )|. Furthermore we call V (f ) := sup {VZ (f ) | Z is a partition of ⌦} the variation of f . We define the space of functions of bounded variation by BV (⌦) := {f : ⌦ ! K | V (f ) < 1} and ||f ||BV := |f (0)| + V (f ), Show that: f 2 BV (⌦). (i) The map ||·||BV : BV (⌦) ! R is a norm and (BV (⌦), ||·||BV ) is a Banach space (ii) The space BV (⌦) with k · ksup is not complete. Hint:(i) You don’t have to show that BV (⌦) is a vector space. Show that ||f ||sup ||f ||BV , f 2 BV (⌦). Exercise 2: Let (||·||n )n2N be a total2 family of semi-norms on a K vector space X. (i) Define for x, y 2 X ✓ ◆ 1 X ||x y||n 1 d(x, y) := . 2n 1 + ||x y||n n=1 Show that d is well-defined and a metric. (ii) Let (xn )n2N be a sequence in X. Show that, xn all k 2 N. n!1 ! x wrt. dif and only if ||xn x||k n!1 ! 0 for (iii) Is the metric d induced by a norm if X 6= {0}? Hint(i) Consider the function f : R Exercise 3: Show that: (i) For 1 p, q 1 with 1 p + 1 q (ii) For 1 p1 , ..., pn 1 with 0 !R 2 7! = 1r , such that r Pn 1 i=1 pi s 1+s . 1, for all x 2 lp (K), y 2 lq (K), we have: kxyklr kxklp kyklq . = 1r , such that r n Y i=1 1 0, s x (i) lr n Y i=1 1, for all x(i) 2 lpi (K), 1 i n, we : kx(i) klpi . A set Z = {x0 , x1 , ..., xn }, n 2 N, is called partition of [a, b] ✓ R, if a = x0 < x1 < ... < xn = b. ||·||n n2N is called total, if ||x||n = 0, 8n 2 N =) x = 0.