Question

a) 13 marks Let C0, 1 be the vector space of all continuous, complex-valued func- tions on the closed interval 0, 1. Define = (If(a)2 dx sup (x) xE[0,1 112 and (i) Show the triangle inequality ||f + g||00 || ||0 ||9||00 (ii) Show that for any function f e C[0, 1, the inequality ||f||2 ||£||2 holds. (iii Show that there exists no fixed constant C such that the inequality SIl0Cf2 holds for allfE C[0, 1 (iv) Construct a sequence {fn} such that it is convergent with respect to ||:||2. norm but it is divergent with respect to ||.||00 norm

Answer 1

13 marks) Let C[0, 1] be the vector space of all continuous, complex-valued func- tions on the closed interval (0,11. Define 1/2 || / || = sup +€10,11 (2) and J Show the triangle inequality / +9|| S ||/\|.c + |19||2c 11) Show that for any function f € C[0,1], the inequality |||||2 < |||||. holds. v Show that there exists no fixed constant C such that the inequality ||$|| <C|||||2 holds for all f e C[0,1]. Viv) Construct a sequence {n} such that it is convergent with respect to 11:12 norm but it is divergent with respect to 1.1 norm. (i) Il figlio = Sup ne [0,1] (f+9) (2) = Sup I fra) + g(x) xe [1] < sup [ 1fCa>H/7621] 26[1] s Sup 26 [0, 1] 1662) + 19 (2) sup ne [0,1] Il fllad + 119 11 "E siecolo sa I fcal da 1.de . If(x) da sup XED]

s 1 scal)" - usato sup (ne[0,1] If (2) | = 1161 Il flle <ifllco (111) if possible , suppose 11f11c <cllfll, - ) I Nel sot VN> then consider the function į Ž i then graph of a would be

(ON) BK – F-12 so that I fill = Sifni calde s fű (a) da da (as fe is positive on [o1]) = area of A = {xw x3 = 1 But clearly Il frolla

So then llfill= rŃ > .c.1= collfall a contradiction to (*) so that ¥ such e. (111) consider fn to be the following then lfullz= Žx nx ? o as n > " "falls = n oo as nya

so then {fn} »o with bespect to Il la norm but divergent winnt llllo normi