Vlll INTRODUCTION
reason, w e have deliberatel y kep t t o a minimum th e use of th e duality theor y vi a
the trace , sinc e w e fee l tha t thi s migh t tur n of f th e reader s wh o ar e no t familia r
with it . Nevertheless, we urge th e readers who want t o go deeper i n th e theor y t o
get acquainte d wit h th e principle s o f thi s dualit y (cf . [P I o r Pe4]). We shoul d
mention tha t ou r restricte d selectio n ha s lef t ou t severa l importan t topics . W e
refer t o [PI ] fo r th e genera l theor y o f operato r ideal s whic h wa s develope d b y
Pietsch an d hi s schoo l since the late sixties. The characterization o f L^-space s (o r
subspaces of L
p
o r subspaces of quotient s of L p) b y operator theoreti c propertie s
is a majo r omission . Fo r this , w e refe r th e reade r t o th e beautifu l pape r o f
Kwapieh [Kw3 ] and t o its references. Also , the factorizatio n theorem s o f Maure y
(and th e importan t wor k o f Rosentha l [R2] ) ar e no t include d here ; w e refe r th e
reader t o [Ml] . W e d o discuss , however , th e genera l theor y o f typ e an d cotype ,
but briefl y an d withou t proofs . W e will be mainly concerne d her e with typ e 2 or
cotype 2 . I n general , w e hav e concentrate d o n th e proble m o f factorin g a n
operator throug h L
2
, an d w e hav e lef t ou t th e natura l extension s fo r th e
factorization throug h L
p
. I n ou r exposition , w e will come acros s most o f th e lin e
of investigatio n which forms th e so-called local theory of Banac h spaces—i.e. , th e
study o f Banac h space s by finite-dimensiona l methods . We have trie d t o indicat e
in th e references , a s ofte n a s possible , th e ramification s o f thi s currentl y ver y
active area .
Let u s no w revie w th e content s o f thes e notes. I n Chapte r 0 , we introduce th e
projective an d injectiv e tenso r products an d th e approximation propert y (i n shor t
A.P.). Amon g th e si x problem s a t th e en d o f th e Resume , th e firs t an d mos t
famous on e wa s th e approximatio n problem : Doe s ever y Banac h spac e posses s
the A.P. ? Enfl o [E ] gave a counterexampl e i n 1972, whic h opene d a ne w er a i n
functional analysis .
In Chapte r 0 , w e hav e insiste d o n th e necessar y distinctio n betwee n nuclea r
operators an d element s o f th e projectiv e tenso r product , whic h i s essentia l i n
Chapter 10.
In Chapte r 1, we present in detail the basic theory of /^-summin g operators an d
its first application s t o Banach spac e theory: Fo r ever y ^-dimensiona l subspac e E
of a spac e X, ther e i s a projectio n P: X - * E suc h tha t ||P| | yfn an d a n
isomorphism T: 1%- E suc h that ||7| |
\\T~l\\
yfn.
In §c , we briefly introduc e ^-integra l operator s an d som e rudiments o f dualit y
theory, bu t thi s i s no t use d i n th e sequel . W e not e i n passin g tha t th e Radon -
Nikodym propert y (whic h is crucial t o compare integral an d nuclea r operators ) is
not discusse d a t al l here; we refer th e reader t o [D-U] for thi s topic. In Chapte r 2,
we give the Lindenstrauss-Pelczyhski criterio n fo r a n operator t o factor throug h a
Hilbert space . This can be viewed as an applicatio n o f th e Hahn-Banach theore m
provided a certain dualit y theore m i s explicited; w e do thi s in §2.b. In Chapte r 3,
we introduc e th e notion s o f typ e an d cotyp e an d prov e Kwapien' s theore m tha t
every space of typ e 2 and o f cotype 2 is isomorphic to a Hilbert space . The theor y
of typ e an d cotyp e provide s a useful scal e to measure ho w clos e a given spac e is
Previous Page Next Page