THE ALGEBRAI C PRELIMINARIE S 3 (in X 0 Y) , then , o f course, ( ] T A ij e i ®/ i )(y?) = 0 for eac h an d ever y ip G -B(X, Y). I n particular , i f x' G X7 an d y' G Y', the n th e bilinear functiona l ip(x', y ') whos e valu e a t (x , y) G X x Y i s given b y P{x,y')(x,y) = x'(x)y'(y) also satisfie s ( YJ X i3ei®f3)({P(x'y'))=0' That is , J2 \i J x'(e l )y'(f j ) = 0 iei,jeJ for eac h x' G X' an d y ' G Y'. Let' s loo k more closel y a t what' s goin g on here. Fo r each x' G X' an d y' G Y' w e have ieijeJ iei,jeJ Since this i s so for eac h x f G X', i t follow s tha t Y2iei,jeJ ^jV'ifj)^ = 0 n e n c e E(EV(/ ))e*=0. and b y £" s linea r independence , w e have o = ^Tx 2j yf(fj) = yf[YlXiifj)' foreach *G J - Since thi s i s so fo r eac h y' G Y', i t follow s tha t X^e j ^u'/ j ~ 0 f°r e a c n * ^ ^ an ^ hence, thank s t o F' s linea r independence , w e have Xij = 0 for eac h j G J an d fo r eac h i G /. By taking a close look at what we've done, one soon sees that th e tensor produc t X' 0 Y ' o f the dua l spaces X' an d Y' ca n be identified wit h a subspace o f 5(X, Y) just loo k a t th e linea r extensio n o f the mappin g x' 0 y' i— if(x',y') t o a ^ ° f X ' 0 Y' . Now that w e have a handle o n X 0 Y , howeve r tenuous , i t i s time t o establis h that X 0 Y doe s the job it wa s created for : linearizing bilinear functions. T o start , consider p G #(X, Y) an d defin e U^ : X 0 Y — K, firs t o n the elementary tensors x 0 y b y U (p (x ^ y) = p(x,y) an d the n exten d i t linearl y t o al l o f X 0 Y . I f w = XoL i £ ® y £ ^ ® Y, the n U^{u) = Y%=i ^(x* Vi) = XX: i ^ ( ^ ® Vi)- U v is easily see n t o b e linea r an d w e have, i n fact , factore d y a s follows : x x y ^ K X 0 Y where th e ma p fro m X x Y int o X 0 Y i s the bilinea r functio n tha t take s th e pai r (x,y) G X x Y t o th e elementar y tenso r x ^ i / G l ® y .

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