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 hi m t o isolat e notion s tha t wer e to pla y a central rol e in the stud y an d
the developmen t o f Banac h spac e theor y t o thi s ver y day .
He was the firs t t o formulat e isomorphi c invariant s o f special Banac h space s b y
comparing thes e space s with othe r Banac h space s vi a the bounde d linea r operator s
between them .
He caugh t an d hel d th e attentio n o f those tha t coul d appreciat e hi s idea s wit h
concrete example s o f endurin g 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 spac e 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 Banac h 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 tha n i n hi s infamou s Resume .
Produced durin g hi s years in Sa o Paulo, th e Resum e set s 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 o f the Resum e wer e demystified an d mad e palatabl e t o a generatio n 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 analysi s an d operato r theory ; a t th e sam e tim e th e
study o f Banac h spac e theor y fo r it s ow n sak e becam e a worthwhil e 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 , the y too k not e o f 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 result s 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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