## First Assignment

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## First Assignment

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- Pages:
- 2
- School:
- Massachusetts Institute of Technology
- Course:
- 18 155 - Differential Analysis

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FIRST ASSIGNMENT DUE SEPTEMBER 11 IN CLASS 18 155 FALL 2001 RICHARD MELROSE In the solutions of these problems I am looking for precise statements and clear succinct proofs Some of these problem may involve things you do not know of course as with the anonymous quiz I am simply trying to check your level of knowledge so that I can adjust the course as necessary No weight in terms of final grade will be given to this assignment but please do it anyway and not anonymously this time Problem 1 Taylor s theorem Let u Rn R be a real valued function which is k times continuously differentiable Prove that there is a polynomial p and a continuous function v such that u x p x v x where lim x 0 v x 0 x k Problem 2 n Let C B be the space of continuous functions on the closed unit ball Bn x Rn x 1 Let C0 Bn C Bn be the subspace of functions which vanish at each point of the boundary and let C Sn 1 be the space of continuous functions on the unit sphere Show that inclusion and restriction to the boundary gives a short exact sequence C0 Bn C Bn C Sn 1 meaning the first map is injective the second is surjective and the image of the first is the null space of the second Problem 3 Measures A measure on the ball is a continuous linear functional C Bn R where continuity is with respect to the supremum norm i e there must be a constant C such that f C sup f x f C Bn x Rn Let M Bn be the linear space of such measures The space M Sn 1 of measures on the sphere is defined similarly Describe an injective map M Sn 1 M Bn Can you define another space so that this can be extended to a short exact sequence 1 2 RICHARD MELROSE Problem 4 Show that the Riemann integral defines a measure Z n 1 C B 3 f 7 f x dx Bn Problem 5 If g C Bn and M Bn show that g M Bn where g f f g for all f C Bn Describe all the measures with the property that xj 0 in M Bn for j 1 n Department of Mathematics Massachusetts Institute of Technology E mail address rbm math mit edu

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