CHAPTER 1 Basics o n tenso r norm s This firs t chapte r present s th e basic s o f tenso r norm s followin g th e spiri t o f Grothendieck's presentation . T o b e sure , we'v e adde d example s an d borrowe d re - sults (wit h proofs ) fro m late r chapter s bu t otherwis e we'v e followe d th e master' s plan. We star t wit h a discussio n o f reasonable crossnorms payin g particula r atten - tion t o tw o examples : th e injectiv e an d projectiv e norms . Thi s i s followe d wit h a discussio n o f som e critica l example s an d Grothendieck' s famou s "computation " of th e dua l o f th e injectiv e tenso r product . A fe w illustrativ e example s follow , i n particular, w e compute th e close d linea r spa n o f (e n ® en)n i i n HP (g £p. Our nex t sectio n i s devote d t o tensor norms define d o n th e tenso r produc t o f finite dimensiona l Banac h spaces . Thi s is followed b y a discussion o f how to exten d the definition o f a tensor norm to the tensor product o f infinite dimensiona l Banac h spaces. Thi s leads to the delicate issues of accessible spaces and tensor norms . Her e we reproduce som e o f Grothendieck' s Memoir t o giv e these notion s som e grit . In th e fourt h section , a-integra l bilinea r form s an d a-integra l linea r operator s make their appearance . I t i s through thes e classes of operators that Grothendieck' s view of Banach space theory becomes clear. Th e fundamental fact s about a-integra l operators include their "ideal " propertie s a s well as their finitary (o r local) determi - nation. Thi s sectio n end s wit h anothe r visi t t o th e delicat e subjec t o f accessibilit y and metri c accessibility . In th e shor t fifth sectio n a-nuclea r form s an d operator s ar e introduce d an d their idea l structur e noted . We clos e thi s chapte r wit h a n expositio n o f Grothendieck' s ofte n overlooke d paper o n th e celebrate d Dvoretsky-Roger s theorem . Th e mai n resul t her e i s hi s generalization o f thei r resul t wit h th e conclusio n tha t i f 1 p o o an d X i s a n infinite dimensiona l Banac h space , the n Crucial t o th e argumentatio n o f this sectio n i s a discussio n o f Blaschke' s selectio n principle, whic h w e include a s a n appendi x t o th e book . The algebrai c preliminarie s Let X , Y an d Z b e linea r space s (ove r th e sam e scala r field K , b e i t th e rea l number field R o r th e comple x numbe r field C) . A functio n tp : X x Y Z i s bilinear i f ip(x, •) : Y Z i s linear fo r eac h x Gl an d £(• , y) : X Z i s linear fo r each y G Y. W e will denote b y B(X, Y\ Z) th e linea r spac e of all bilinear function s from X x Y t o Z an d b y B(X, Y) th e spac e of all bilinear function s o n l x Y int o the scala r field. 1 http://dx.doi.org/10.1090/mbk/052/01
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