Chapter 1 The Cauchy-Schwar z inequality In this section we shall follow a procedure often considere d nasty , b the on e I hop e t o convinc e yo u t o appreciate . W e shal l wor k back wards, discoverin g concept s a s w e g o along , instea d o f statin g the ahead o f time. N o background beyon d hig h school mathematics i s r quired t o rea d thi s chapter . Fo r furthe r informatio n o n th e materi presented here , the reade r i s encouraged t o tak e a look a t J . Micha Steele's beautifu l book , entitle d The Cauchy-Schwarz Master Clas ([16])- A quic k perspectiv e befor e w e begi n th e nut s an d bolt s o f th mathematical discussion . Inequalitie s ar e a dime a dozen. Th e stat ment tha t 2 3 is a true inequality , bu t i t i s meaningless. A n inte esting inequalit y i s one tha t come s clos e t o no t bein g true , bu t doe not quit e cros s tha t precariou s threshold . Thi s i s not muc h differen than an y area of learning. Sayin g that th e United State s has a bigg per capita income than Cha d is a true but meaningles s statement. O the other hand, th e fact tha t th e People's Republic of Congo has lan area equa l t o nearl y tw o thirds o f the lan d are a o f the Unite d State is muc h mor e precise , interestin g an d surprising . T o pu t i t bluntl to sa y somethin g interesting , on e mus t wal k o n th e ver y edg e o f th
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