322finsoln

322finsoln - B U Department of Mathematics Spring 2006 Math...

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Unformatted text preview: B U Department of Mathematics Spring 2006 Math 332 - Real Analysis II, Final, 29/5/2006, 15:00-17:30 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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322finsoln - B U Department of Mathematics Spring 2006 Math...

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