Preface It's bee n mor e tha n hal f a centur y sinc e Alexande r Grothendiec k burs t ont o the mathematica l scene . Hi s natura l gif t fo r ap t abstrac t generalisation s wa s first tested i n th e aren a o f functiona l analysi s an d wa s no t foun d wanting . Hi s inbor n compass le d him to isolate notions that wer e to play a central role in the study an d the developmen t o f Banach spac e theory t o thi s ver y day . He was the first t o formulate isomorphi c invariant s o f special Banach space s by comparing thes e space s with othe r Banac h space s via the bounde d linea r operator s between them . He caught an d hel d th e attentio n o f those tha t coul d appreciat e hi s ideas wit h concrete example s o f enduring importance . He recognize d th e importanc e o f th e natur e an d locatio n o f th e finite dimen - sional subspace s o f a space an d utilize d suc h "local " theor y wa s born . He wa s th e first analys t t o seriousl y chas e diagram s i n th e hope s o f catch - ing essentia l isomorphi c characteristic s o f Banach spaces , an d catc h the m h e mos t certainly did . Nowhere ar e thes e innovation s mor e i n evidenc e than i n his infamous Resume . Produced durin g his years in Sao Paulo, the Resume sets forth Grothendieck' s pla n for th e stud y o f th e finer structur e o f Banac h spaces . H e use s tenso r product s a s a foundatio n upo n whic h h e build s th e classe s o f operator s mos t importan t t o th e study an d establishe s th e importanc e o f th e "local " theor y i n th e stud y o f thes e operators an d th e space s the y ac t upon . Whe n i n th e lat e sixties , Jora m Lin - denstrauss an d Aleksande r Pelczynsk i redresse d hi s Fundamental Inequalit y i n th e trappings o f operato r ideals , i t signale d th e rebirt h o f Banac h spac e theory . Th e ideas of the Resume were demystified an d mad e palatabl e to a generation o f math - ematical analysts . Banac h spac e theor y soo n attracte d a sle w o f talente d youn g mathematicians who , wit h Lindenstraus s an d Pelczynsk i a t th e lead , establishe d the subjec t a s a worth y ai d i n studyin g th e problem s o f mor e classica l aspect s o f mathematical endeavor , suc h a s harmoni c analysis , probability , comple x analysis , geometry o f convex bodies, rea l analysis an d operato r theory a t th e sam e time th e study o f Banach spac e theor y fo r it s ow n sak e became a worthwhile occupation . To b e sure , muc h o f th e succes s o f th e wor k o f Lindenstraus s an d Pelczynsk i is du e t o thei r sheddin g Grothendieck' s Fundamenta l Inequalit y o f it s mystifyin g tensorial formulation . Nevertheless , they too k not e of what the y ha d done . T o wit, "Though th e theor y o f tenso r product s constructe d i n Grothendieck' s pape r ha s its intrinsi c beaut y w e fee l tha t th e result s o f Grothendiec k an d thei r corollarie s can b e mor e clearl y presente d withou t th e us e o f tenso r products . Th e pape r o f Grothendieck i s quite har d t o rea d an d it s results ar e no t generall y know n eve n t o experts i n Banac h spac e theory. " vii
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