18. The Jacobian Formula - Probability

tutorial 18 : The Jacobian Formula118. The Jacobian FormulaIn the following, 125We callK-normed space,anorderedpair(E, N),whereEis aK-vector space, andN:E R+is a norm definition (89)forvector space, and definition (95) , be an inner-product on aK-vector Show that = , is a norm Show that (H, )isaK-normed (E, )beaK-normed space:1. Show thatd(x, y)= x y defines a metric Show that for allx, y E,wehave| x y | x y . 18: The Jacobian Formula2 Definition 126 Let(E, )be aK-normed space, anddbe themetric defined byd(x, y)= x y .Wecallnorm topologyonE,denotedT , the topology onEassociated that this definition is consistent with definition (82) of the normtopology associated with an , Fbe twoK-normed spaces, andl:E Fbe alinear map. Show that the following are equivalent:(i)lis continuous ( to the norm topologies)(ii)lis continuous atx=0.(iii) K R+, x E, l(x) K x (iv)sup{ l(x) :x E, x =1}<+ Definition 127 LetE,FbeK-normed spaces.

Tutorial 18: The Jacobian Formula 1 18. The Jacobian Formula In the following, K denotes R or C. Definition 125 We call K-normed space,anorderedpair(E,N), where E is a K-vector space, and N: E → R+ is a norm on E. See definition (89)forvector space, and definition (95)fornorm. Exercise 1.

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