viii PREFAC E What the y sai d i n the lat e sixtie s stil l applies ! However , w e think th e Resum e still has much to offer an d we believe that it s contents are still worthy of close study it i s to support suc h a view that w e devote thi s work. Mainl y w e have followed th e path blaze d b y Grothendieck w e have presente d mos t o f th e argument s usin g th e machinery availabl e to him at th e time of discovery. T o be sure, Grothendieck kne w what mattere d t o Banac h spac e affair s i n grea t detai l an d use d hi s geniu s t o pu t much of the best o f that materia l to work efficiently an d effectivel y i n the executio n of his plan. Ther e ar e several junctures wher e "moder n technology " migh t shorte n some arguments usuall y w e relegate suc h t o ou r Note s an d Remarks . To ensur e a clea r understandin g o f just wha t tool s wer e availabl e i n Grothen - dieck's functiona l analysi s days , w e hav e include d severa l appendices . Thoug h w e sometimes op t fo r a more modern presentatio n tha n wa s available i n the fifties , w e stay faithfu l t o th e formulatio n o f the result s a s use d b y Grothendieck . A brief outlin e o f the content s follows . In Chapte r 1 , we present i n detail th e basi c facts an d feature s o f tensor norms , including ho w th e integra l bilinea r form s an d operator s deriv e fro m a given tenso r norm. Thi s chapter has been presented to a number of defenseless graduate student s throughout th e year s an d th e leve l of detai l i s a consequence . Recal l th e word s of Professor C . A. Rogers, who in the Introduction t o his treasured book on "Hausdorf f Measures", acknowledge s tha t hi s "boo k i s largely base d o n lectures , and , a s I like my students to follow my lectures, proofs are given in great detail this may bore the mature mathematician , bu t I believe will be a great hel p t o anyon e tryin g t o lear n the subjec t a b initio. " W e hav e take n Professo r Rogers ' word s a s soun d advice , particularly i n ligh t o f the ari d natur e o f the initia l aspect s o f tensor norms . Chapter 1 , then , i s devote d t o th e stud y o f tenso r norm s an d th e operato r ideals generate d b y them . W e hav e inserte d examples , a s wel l a s som e simpl e computations, tha t w e believ e "wil l eas e th e pain " a bit . Again , wit h a n ey e t o exposing th e seriou s studen t t o ho w th e mos t classica l tenso r norm s (th e projec - tive an d injectiv e norms ) behave , w e hav e ende d Chapte r 1 with a n expositio n o f Grothendieck's treatmen t o f th e Dvoretsky-Roger s theorem . Her e w e se e ho w i n infinite dimensiona l Banac h space s tha t th e collection s o f absolutel y p-summabl e series an d weakl y p-summabl e serie s constitut e vastl y differen t collection s whe n p is a real number large r tha n o r equal to 1 , we compare thes e collection s to th e pro - jective an d injectiv e tenso r product s o f the classica l sequence space s with a genera l Banach spac e o f infinitel y man y dimensions . In Chapte r 2 the centra l rol e playe d b y C-space s an d th e L-space s i n Banac h space theor y i s firml y established . First , w e investigat e integra l operators , thei r remarkable characterizatio n i n terms o f factorization, an d th e relationship betwee n the differentiabilit y o f vector-value d measure s an d th e nuclearit y o f certai n oper - ators actin g o n Lebesgu e spaces . Alon g th e wa y w e fin d tha t integra l operator s into L-spaces ar e precisely thos e that tak e the close d uni t bal l of their domai n int o an orde r bounde d set . W e build o n thes e peculiaritie s (an d associate d phenomen a in C-spaces ) t o provid e a platfor m fo r th e discussio n o f injectiv e an d projectiv e tensor norms her e th e seemingl y ari d wastelan d o f Chapte r 1 springs t o lif e an d we ar e rewarde d fo r ou r carefu l wor k b y th e shar p characterization s o f left , right , and two-side d injective/projectiv e tenso r norm s i n term s o f integral form s an d op - erators. W e then appl y thi s theor y t o loo k a t ho w variou s injectiv e an d projectiv e tensor norm s ar e derive d fro m a given tenso r nor m an d tak e particula r pleasur e i n
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