4 1. BASIC S O N TENSO R NORM S On the othe r hand , i f we start wit h a linear functiona l U on X 0 Y an d defin e (pu on X x Y b y the formul a Vu{x,y) = U(x®y), then i t i s plai n an d eas y t o se e tha t /?£ / is bilinear , thank s i n larg e par t t o th e already note d bilinearit y o f x 0 y. If we step bac k an d tak e stoc k o f what's bee n done , w e should realiz e tha t startin g wit h th e bilinea r ip and passin g t o th e linea r U^ , i f we now loo k at (pu , we're bac k a t (p startin g wit h th e linea r U and passin g t o th e bilinea r (pu, if we next loo k at U (pu , we'r e bac k a t U. In othe r words , B(X,Y) an d 1 0 7 , ar e naturall y isomorphi c wit h th e diagra m X xY telling th e whol e story . Thi s stor y i s ofte n calle d th e universal mapping property of tensor products . It i s noteworth y tha t th e fac t tha t th e linea r and/o r bilinea r function s too k values i n th e scala r fiel d K wa s unimportan t t o th e argument . I n fact , i f Z i s an y linear space (ove r the same field a s X an d Y), then the Universal Mapping Propert y has a byproduct th e diagra m X xY establishes a natural isomorphis m betwee n B(X, Y Z) an d L(X x Y Z) wit h root s in th e formul a U(x®y) = (p(x,y). Lest ther e b e concer n o f precisel y wha t w e hav e constructed , b e assure d tha t the tenso r produc t we'v e buil t i s uniquel y qualifie d t o no t onl y d o th e jo b w e se t for i t (t o linearize bilinear functionals) , bu t mor e so, to linearize bilinear operators . Indeed w e have th e following : THEOREM. Let X and Y be linear spaces and let W be a linear space and r : X xY Z be a bilinear map with the property that for any linear space Z and any bilinear function (p : X xY Z, there is a unique linear function L :W Z such that ^p Lor. Then there is a linear isomorphism J : X (g ) Y W such that J(x 0 y) = T(X, y) for each x £ X,y eY. In other words , X 0 Y i s unique to the exten t o f being abl e to linearize bilinea r maps wit h minima l mus s an d fuss .
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