2 1. Th e Cauchy-Sehwar z inequali cliff o f falsehood , ye t neve r fal l off . I t i s tim e t o begi n th e perilo journey. Let a and b denote tw o real numbers . The n (a - bf 0. This statemen t i s so vacuous, yo u ar e probabl y wonderin g wh am tellin g you this . Nevertheless , expan d th e lef t han d t o se e tha a2 - 2ab + b 2 0, which implie s tha t (i.D * ^ . Now conside r N AN = ^a k ax N BN = ^ ^f c = h + • + a-N, 6iv, fc=i where a i , . . . , a/v, an d &i,... , 6JV are real numbers . Le t FiV / N Our goa l is to tak e advantag e o f (1.1) . Let' s tak e a look a t N N 7 , ®kbk = XNYN 2^ (1.2) fc = l fc = l iV X N Y N J2 k=i Xiv YN Exercise 1.1 . Explai n wh y i f C i s a constant , the n N N ^2Ca k =C^2a k . fc=i fe=i

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