2 1. BASIC S O N TENSO R NORM S We will denot e b y X' th e algebraic dua l o f X, that is , the linear spac e o f all linear functional s o n X. L(X Z) will denote the linear spac e of all linear function s from X t o Z. The basi c questio n answere d b y tensor produc t construction s i s the following: Is ther e a linea r spac e V suc h tha t L(V Z) coincide s wit h (i s isomorphic to , i s naturally isomorphi c to ) B(X, Y Z)7 Rephrasin g th e question: Ca n we in some way lineariz e bilinea r functions ? The answe r i s "yes" an d the objec t w e construct, th e tensor product , X 0 Y, of X an d Y wil l do the job. Since we are looking for a vector spac e V which, in particular, ha s a dual tha t is (isomorphi c to ) the space B(X, Y)' o f bilinear functionals , i t is natural t o look inside th e dual B(X, Y)' o f B(X,Y). Therein , w e find a collection o f functional s of the for m x 0 y, where x £ X an d y EY: X 0 y (calle d an elementary tensor ) i s the elemen t o f B(X, Y)' whos e value at p G B(X, Y) i s given by the evaluation (x0y)((/?) = ( P(xy)' The tensor product X 0 Y is the linear spa n of the collection of elementary tensors , {x ®y : x £ X,y e Y}. S o a typical u G X 0 Y ha s the form n (*) U = ^ ^ ^i x i ® V% i=l where Ai,... , An ar e scalars, x\, ..., x n G X an d yi,..., yn G Y an d n G N i s arbitrary. The behavio r of X 0 Y i s worth emphasizing , x 0 y has itself a certain degre e of bilinearity. Here' s what' s so: (1) {xi +x2)®y = (xi ® y) + (x2 0 y), (2) x (g) (yi + V2) = (x® 2/i) + (x (g) y2), (3) A( x ®y) = \xgy = Ay, (4) 0 0 y = x(g)0 = 0 x ®y. The reaso n wh y these relation s ar e true i s because o f the way we are forced t o determine whe n u, v G X 0 Y ar e the same: u = v precisely whe n u(ip) = v(ip) for each ip G B(X,Y). I t is also obviou s fro m thes e relation s tha t th e representation (*) o f u G X 0 Y i s far from unique . All we've said so far is easy enoug h to verify an d soon lead s to more insightfu l features o f life insid e X 0 Y. Here' s one : Le t E be a linearly independen t subse t of X an d let F be a linearly independen t subse t o f Y, the n th e set E®F :={e®f :eeE,f eF} is linearl y independen t i n X 0 Y. Indeed , loo k a t any finite linea r combinatio n of element s o f E 0 F. Withou t los s of generality, w e may assume tha t thi s linea r combination i s of the form ieijeJ where the sets I an d J ar e finite subsets of E and F respectively. No w if ^ A^e 0 / ^ =0 i£l,j£j
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