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Unformatted text preview: B U Department of Mathematics Spring 2006 Math 332  Real Analysis II, Final, 29/5/2006, 15:0017:30 Whatever theorem, proposition, lemma you use, Full Name : you must be sure that it is applicable. Student ID : More hint means more detail is required! over 50 1. [4] Suppose f, g : R n → R k are smooth functions; x , x ∈ R n . In the statements below, fill in the blanks to make the claims meaningful (not necessarily true) and to make precise what norm is meant: (a)  f ( x ) f ( x ) Df ( x )( x x )  ......  x x  ...... can be made arbitrarily small. (b)  Df  ...... is bounded. (c)  D 2 f ( x )( x, y )  ...... is not necessarily bounded. (d)  f g  ...... = sup x { f ( x ) g ( x )  ...... } . 2. We had remarked that D 2 f is bilinear. Is it linear? Below you are going to answer that, and more. Consider a function T : R a × R b → R . For each statement below, if your answer is yes, prove; if no, give a counterexample (a) [3] If T is linear, is it true that it is always bilinear with respect to R a and R b ?...
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 Spring '11
 talinbudak
 Math, Vector Space, Continuous function, Inverse function, Inverse Function Theorem, Contraction mapping

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