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 comput e 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 i s followed b y a discussio n o f how to exten d
the definitio n o f a tensor nor m t o the tensor produc t o f infinite dimensiona l Banac h
spaces. Thi s leads to the delicate issue s of accessible space s an d tenso r norms . Her e
we reproduce som e o f Grothendieck' s Memoir t o giv e thes e 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 classe s of operators tha t Grothendieck' s
view of Banach space theory becomes clear. Th e fundamental fact s abou t a-integra l
operators includ e 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 thi s sectio n i s a discussio n o f Blaschke' s selectio n
principle, whic h w e includ e 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 G l 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 o f al l bilinear function s
from X x Y t o Z an d b y B(X, Y) th e spac e o f al l bilinea r function s o n l x Y int o
the scala r field.
1
http://dx.doi.org/10.1090/mbk/052/01
Previous Page Next Page