4 1. Th e Cauchy-Schwar z inequalit Following th e proof o f Cauchy-Schwart z abov e i t i s not difficu to see that i t suffice s t o prove tha t (1-8) a fc- + V p p f Exercise 1.4 . Chec k the details of this and demonstrate lin e by li that Holder' s inequait y indee d follow s fro m (1.8) ! To prov e (1.8 ) we need t o use first yea r calculus . I f you are n familiar wit h calculus , tak e (1.8 ) for grante d an d move fo r now. D not worry.. . If you are familiar wit h basi c calculus , se t a p = ex an d If = e Let - = t an d observ e tha t 0 t 1 . W e ar e the n reduce d p _ showing tha t fo r any real value d x, y and t G [0,1], e b-t)*+tv (l-t)e x + te y . Let Fit) = ( 1 - t)e x + te y - e ^^x+ty and observ e tha t F(0 ) = F(l ) = 0 . Als o observe tha t F"(t) = -~(x - y) 2 €{1~t)x+ty 0. What d o we have? A twice differentiabl e functio n F i s equal 0 a t f = 0 and t = 1 . Moreover , F i s concave u p because F ,f (t) We ar e force d t o conclud e tha t F i s alway s non-negativ e an d (1. follows. Exercise 1.5 . Prov e the Cauchy-Schwartz inequalit y in the followi way. Le t a = (ai,..., ajsr) an d b = (&i,... , . Defin e a , b = ai&i + h djv^Ar, and defin e II I I 2 2 i 2 I i 2 ||a|| = a 1 -ha2~\ \~a N . Cauchy-Schwarts take s the for m a,6|H|-[|6||. Consider a tb,a tb , expand i t out, write it as a quadr polynomial i n t, minimiz e i t in t, an d complete th e proof.
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