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.