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FUNCTIONAL ANALYSIS: NOTES AND are the notes prepared for the course MTH 405 tobe offered to graduate students at IIT Basic Inequalities12. Normed Linear Spaces: Examples33. Normed Linear Spaces: Elementary Properties54. Complete Normed Linear Spaces65. Various Notions of Basis96. Bounded Linear Transformations157. Three Basic Facts in Functional Analysis178. The Hahn-Banach Extension Theorem209. Dual Spaces2310. Weak Convergence and Eberlein s Theorem2511. Weak* Convergence and Banach s Theorem2812. Spectral Theorem for Compact InequalitiesExercise :(AM-GM Inequality) Consider the setAn={x= (x1, ,xn) Rn:x1+ +xn=n, xi 0 everyi},and the functiong:Rn R+given byg(x1, ,xn) =x1 :(1)Anis a compact subset ofRn,andgis a continuous function.(2) Letz= (z1, ,zn) Rnbe such that maxx Ang(x) =g(z).Thenzi= 1 for alli.(Hint. Letzp= minziandzq= maxzifor some 1 p,q (y1, ,yn) Anbyyp= (zp+zq)/2 =yqandyi=zifori6=p, < zqtheng(y)> g(z).)
FUNCTIONAL ANALYSIS: NOTES AND PROBLEMS 3 Exercise 1.9 : (H older’s Inequality for measurable functions) Let p;q>1 be conjugate exponents. Let f and g be Lebesgue measurable complex-
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