INDEX O F NOTATIO N 275 7i(X,Y) Th e collectio n o f al l continuou s bilinea r functional s o n 11 2 X x Y satisfyin g th e equivalen t condition s se t fort h i n Proposition 3.1. 1 Tensor product s x g y A n elementar y tenso r 2 t (g 8)h) Th e transposition of the tensor g®h unde r the transposition 2 6 map X 0 Y Th e algebrai c tenso r produc t o f the vecto r space s X an d Y 2 (X (g ) Y, a) Th e tensor product X®Y assigne d with a reasonable cross - 5 norm a a. X g ) Y Th e completio n o f X 0 Y equippe d wit h th e nor m a 7 a* Th e dua l nor m associate d wit h th e tenso r nor m a 2 7 t a Th e transpos e o f the tenso r nor m a 2 6 v a Th e contragradien t nor m associate d wit h th e tenso r nor m a: a o f a give n b y a= f (a*) (*a)* | |v o r V Th e injectiv e tenso r nor m 7 v X ®Y Th e completio n o f X ® F wit h respec t t o | | v , the injectiv e 1 0 tensor produc t o f X an d F | | A o r A Th e projectiv e tenso r nor m 7 A X (g ) Y Th e completion of X®Y wit h respect t o | |A, the projective 1 0 tensor produc t o f X an d Y /a Th e lef t injectiv e hul l o f a 9 0 a\ Th e righ t injectiv e hul l o f a 9 0 \a Th e lef t projectiv e hul l o f a 9 0 a/ Th e righ t projectiv e hul l o f a 9 0 H or | |H Th e Hilbertia n tenso r nor m 11 6 H* or | |H* Th e dua l Hilbertia n tenso r nor m 11 3 KG Grothendieck' s constan t 15 2 Compact an d conve x set s %(S) Th e collection of all non-empty compact subset s of the com- 21 1 pact metri c spac e S D(Ki,K2) Th e Hausdorf f distanc e betwee n tw o compac t set s Ki an d 21 1 K2 £(JKTI, K2) Th e distanc e between tw o compact set s K\ an d K 2 wr t th e 21 2 metric 5 Cn Th e collectio n o f al l non-empt y compac t conve x subset s o f 21 2 the close d uni t bal l Bp o f R n SK(-) Th e suppor t functio n o f K £ 6 n 21 4 vol(K) Th e volum e functio n vo l : Cn - R + 21 3 £(F) Th e collectio n o f al l ellipsoid s containe d i n th e close d uni t 21 6 ball Bp o f a finite dimensiona l Banac h spac e F Banach lattice s x V y Th e leas t uppe r boun d o f x an d y i n a n ordere d spac e x A y Th e greates t lowe r boun d o f x an d y i n a n ordere d spac e
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