8. Problems for boundede operators 1. Let A: (2 → l² be given by 3 n+1 A(x1,., xn, ...) = In.. 2,.... n+ 2 a) Find the norm of A. b) Find the adjoint A'. c) Find the norm of A'. 2. Let A : L (0, 1) →...

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Extracted text: 8. Problems for boundede operators 1. Let A: (2 → l² be given by 3 n+1 A(x1,., xn, ...) = In.. 2,.... n+ 2 a) Find the norm of A. b) Find the adjoint A'. c) Find the norm of A'. 2. Let A : L (0, 1) → L²(0, 1) be given by (Af)(z) = (r+t)f(t)dt. Find the adjoint A'. 3. Let K(x, y) = xy(x+ y), 0 < x,="" y="">< 1="" and="" consider="" the="" following="" operator="" k="" k="" f(y)="|" k(y,="" x)f(x)dx.="" a)="" is="" this="" operator="" bounded="" as="" an="" operator="" acting="" from="" l'(0,="" 1)="" into="" itself?="" b)="" is="" k="" bounded="" as="" an="" operator="" acting="" from="" l(0,="" 1)="" into="" l'(0,="" 1)?="" c)="" is="" k="" bounded="" as="" an="" operator="" acting="" on="" l²(0,="" 1)?="" 4.="" consider="" the="" operator="" a:="" l²(0,="" 1)="" →="" l²(0,="" 1)="" from="" problem="" 2.="" a)="" is="" this="" operator="" bounded?="" if="" "yes",="" find="" it's="" norm.="" b)="" is="" the="" open="" mapping="" theorem="" applicable="" to="" this="" operator?="">