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
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