PREFACE i x pursuing th e natura l tenso r norms . W e close Chapte r 2 with a partial tabl e o f th e natural tenso r norms , comparin g thos e thu s fa r encountered . In Chapte r 3 , Hilbert space s join in the fun. After establishin g th e existenc e of the Hilbertian tensor norm H and proving that th e H-integral operators between two Banach spaces are precisely those that facto r throug h a Hilbert space , H is shown to be injective an d s o is easily comparable t o the injective hul l of the projective tenso r norm, the so-called pre-integral norm. W e briefly investigat e the hermitian H-form s and thei r compadre s th e hermitia n H*-forms surprisin g measure-theoreti c conse - quences ar e drawn . Thi s i s followe d wit h th e proo f o f wha t i s no w know n a s "th e little Grothendiec k theorem" , whic h has as a corollary th e fact tha t o n the produc t of C-space s th e H-form s an d H*-form s coincide . W e clos e the chapte r relatin g th e various classes of integral operators relativ e to the natura l tenso r norm s wit h idea s from th e classical theory of operators betwee n Hilbert spaces . Th e Hilbert-Schmid t operators ar e show n t o coincid e wit h thos e operator s betwee n Hilber t space s tha t factor throug h a n L-space alternativel y the y ar e show n t o b e precisel y thos e tha t factor throug h a C-space . A remarkabl e consequenc e o f th e loca l characte r o f th e classes o f integral operator s appear s herein : Ever y Hilber t spac e i s simultaneousl y isomorphic t o a subspac e o f a n L-spac e an d a quotien t o f a C-space . I t i s als o shown tha t ever y operato r fro m a n L-spac e int o a Hilber t spac e ca n b e extende d to an y large r domai n i n a continuou s linea r fashion . Chapter 4 i s wher e th e fundamenta l theore m o f th e metri c theor y o f tenso r products i s first formulated . Followin g Grothendieck , w e giv e a numbe r o f it s consequences a s well as his origina l proof . Throughout th e tex t w e have included a number o f Notes an d Remark s which , we hope , roun d ou t th e presentation . W e hav e trie d t o sta y t o th e poin t an d that is , a s w e se e it , t o expos e th e Resume . Followin g th e mai n text , w e hav e four appendices . Th e first discusse s th e solution s t o th e ope n problem s liste d b y Grothendieck. Sinc e the solution s t o thes e problem s involve d notion s tha t evolve d later tha n th e appearanc e o f th e Resume , w e hav e include d a ver y brie f Glossar y of terms . Wit h thes e term s i n hand , th e discussio n o f th e problem s ough t t o b e sufficient t o give an overview of the disposition of these problems and their solutions. There follo w thre e mor e appendice s wherei n w e discus s result s tha t Grothen - dieck use d tha t ar e critica l t o th e understandin g o f the Resum e bu t ar e somewha t scattered acros s the literature. Her e we bow to the go d o f convenience, usin g mod - ern exposition s i n order t o eas e th e pai n somewhat . We woul d b e remis s i f w e di d no t acknowledg e th e wor k o f man y other s o n tensor product s an d operator ideal s that influence d ou r work. Th e aler t reade r wil l find commentary o n the works of these mathematician s i n our Notes and Remarks, which ar e scattere d throughou t thes e deliberations . To be sur e w e make specia l mentio n her e o f the work s of Amemiy a an d Shig a (1957), Lindenstraus s an d Pelczyhsk i (1968) , Gilber t an d Lei h (1980) , Pietsc h (1980), an d Defan t an d Flore t (1993) . Eac h ha s clarifie d fo r u s man y o f th e mysteries encountere d i n th e Resume. A recen t additio n t o th e literatur e o n tenso r product s i s th e charmin g boo k by RayRya n (2002 ) "Introductio n t o Tenso r Product s o f Banac h Spaces" thos e who ar e neophyte s i n tenso r product s wil l find Ryan' s treatmen t sympatheti c an d comforting.
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