INTRODUCTION

IX

from a Hilbert space . We briefly revie w the main points of thi s theory in §3.3 (we

use onl y th e extrem e cases of typ e 2 or cotyp e 2 in th e sequel) . In Chapte r 4 , we

prove a factorization theore m which links Kwapieh's theore m and Grothendieck' s

theorem: I f X* an d Y ar e of cotype 2, then every approximable operato r fro m X

into Y factor s throug h a Hilber t space . Thi s resul t play s a crucia l rol e i n th e

construction o f Chapte r 10. As an application, in §4.b, we show that Sido n set s in

the dua l o f a compac t Abelia n grou p G ar e characterize d b y th e fac t tha t the y

span a cotyp e 2 spac e i n C(G). Thi s generalize s a n earlie r resul t o f Varopoulo s

[VI]. I n Chapte r 5 , w e concentrat e o n Grothendieck' s theorem , whic h w e ab -

breviate G.T. Chapte r 5 contains at least fou r proof s o f tha t theorem . In §5.a, we

briefly introduc e ^-space s (ther e is more information i n §§8. b an d 8.c) . We ar e

mainly concerne d her e with th e case s p = 1 and p = oo . This allow s us t o stat e

and prove G.T. in the framework o f [L-P]: Every operator fro m a n 2£

x

spac e into

an ££

2

spac e is 1-summing. Thi s is proved i n §5.c. In §5.b we give the (somewha t

dual) formulatio n abou t operator s define d o n a C(AT)-spac e or o n a n o^-space .

We trie d t o give explicitly all the various forms in which the theorem can be used,

and w e distinguishe d carefull y betwee n th e eas y par t (whic h w e cal l th e "littl e

G.T.") an d th e mor e delicat e part o f thi s theorem . We firs t giv e a proof derive d

from th e mor e "abstract " resul t o f Chapte r 4 , bu t §5. d contain s anothe r proof ,

more direct and o f independent interest .

In §5.3 , we include Krivine' s proo f o f G.T. , whic h give s the best know n uppe r

bound fo r th e constant K G. I n problem 3 in the Resume, Grothendieck aske d fo r

the exac t valu e o f variou s constants (se e 5.3 fo r details) ; thi s is the only proble m

which i s no t completel y solve d (bu t o f course , i t i s probably th e leas t importan t

one!). In 5.f , w e give a very quick proof o f G.T., based o n a property o f th e spac e

Hl9 du e to Pelczyhski and Wojtaszczyk .

In Chapte r 6 , w e stud y th e Banac h space s satisfyin g G.T. , whic h w e call G.T .

spaces. We include several characterizations of thes e spaces, but we insist more on

the a prior i smalle r clas s o f G.T . space s o f cotyp e 2 . Th e latte r enjoy s nice r

stability properties and includes all the known examples of G.T. spaces. In 6.c, we

show that if R i s a hilbertian (or, more generally, a reflexive) subspac e of L

l9

the n

LY/R i s a G.T. space of cotype 2.

We com e her e t o proble m 5 i n th e Resume . A stronge r formulatio n o f thi s

problem wa s give n b y Lindenstraus s an d Pelczyhski , wh o aske d whethe r th e

^-spaces ar e th e onl y space s satisfyin g G.T . Th e abov e resul t o f 6. c (du e t o

Kisliakov an d th e author) gives a negative answer t o thi s question (an d a fortior i

to problem 5 ) since the quotients L

x

/R ar e never ^-spaces whe n R i s a reflexiv e

infinite-dimensional subspac e of L

v

It i s rathe r eas y t o giv e concret e example s o f hilbertia n subspace s o f L x: fo r

example, th e spa n o f th e Rademache r functions . However , i t i s more delicat e t o

produce " very large " suc h spaces . Fo r thi s purpose , w e presen t i n Chapte r 7 a

method base d o n volum e estimates whic h yields a n orthogona l decompositio n o f

l\n int o tw o parts whic h ar e uniformly (wit h respect t o n) isomorphi c t o /£ . This