FIRST ASSIGNMENT, DUE SEPTEMBER 11 IN CLASS18.155 FALL 2001RICHARD MELROSEIn the solutions of these problems I am looking for precise state ments 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 levelof knowledge so that I can adjust the course as necessary. No weight, in termsof final grade, will be given to this assignment but please do it anyway and notanonymously this time.Problem 1[Taylor’s theorem]. Let u : Rn−→ R be a real-valued function which is k timescontinuously differentiable. Prove that there is a polynomial p and a continuousfunction v such thatu(x) = p(x) + v(x) where lim|x|↓0|v(x)||x|k= 0.Problem 2Let C(Bn) 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 vanishat each p oint of the boundary and let C(Sn−1) be the space of continuous functionson the unit s phere. Show that inclusion and restriction to the boundary gives ashort exact sequenceC0(Bn) ,→ C(Bn) −→ C(Sn−1)(meaning the first map is injective, the second is surjective and the image of thefirst is the null space of the second.)Problem 3[Measures] A measure on the ball is a continuous linear functional µ : C(Bn) −→R where continuity is with respect to the supremum norm, i.e. there must be aconstant C such that|µ(f)| ≤ C supx∈Rn|f(x)| ∀ f ∈ C(Bn).Let M(Bn) be the linear space of such measures. The space M (Sn−1) of measureson the sphere is defined similarly. Describe an injective mapM(Sn−1) −→ M(Bn).Can you define another space so that this can be extended to a short exact sequence?12 RICHARD MELROSEProblem 4Show that the Riemann integral defines a measure(1) C(Bn) 3 f 7−→ZBnf(x)dx.Problem 5If g ∈ C(Bn) and µ ∈ M (Bn) show that gµ ∈ M (Bn) where (gµ)(f ) = µ(f g) forall f ∈ C(Bn). Describe all the measures with the property thatxjµ = 0 in M (Bn) for j = 1, . . . , n.Department of Mathematics, Massachusetts Institute of Tech nologyE-mail address:
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