MIT OpenCourseWarehttp://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � Problem set 7, Due 11AM Tuesday 7 Apr. I will put up some practice problems for the test next Thursday when I get a chance. Problem 7.1 Question:- Is it possible to show the completeness of the Fourier basis exp(ikx)/√2π by computation? Maybe, see what you think. These questions are also intended to get you to say things clearly. (1) Work out the Fourier coefficients ck(t) = (0,2π) fte−ikx of the step function (14.25) ft(x) = 1 0 ≤ x < t 0 t ≤ x ≤ 2π for each fixed t ∈ (0, 2π). (2) Explain why this Fourier series converges to ft in L2(0, 2π) if and only if (14.26) 2 |ck(t)|2 = 2πt − t2 , t ∈ (0, 2π). k>0 (3) Write this condition out as a Fourier series and apply the argument again to show that the completeness of the Fourier basis implies identities for the sum of k−2 and k−4 . (4) Can you explain how reversing the argument, that knowledge of the sums of these two series should imply the completeness of the Fourier basis? There is a serious subtlety in this argument, and you get full marks for spotting it, without going ahead a using it to prove completeness. Problem 7.2 Prove that for appropriate constants dk, the functions dk sin(kx/2), k ∈ N, form an orthonormal basis for L2(0, 2π). Hint: The usual method is to use the basic result from class plus translation and rescaling to show that d�exp(ikx/2) k ∈ Z form an orthonormal basis of k L2(−2π, 2π). Then extend functions as odd from (0, 2π) to (−2π, 2π). Problem 7.3 Let ek, k ∈ N, be an orthonormal basis in a separable Hilbert space, H. Show that there is a uniquely defined bounded linear operator S : H −→ H, satisfying (14.27) Sej = ej+1 ∀ j ∈ N. Show that if B : H −→ H is a bounded linear operator then S +�B is not invertible if � < �0 for some �0 > 0. Hint:- Consider the linear functional L : H −→ C, Lu = (Bu, e1). Show that B�u = Bu − (Lu)e1 is a bounded linear operator from H to the Hilbert space91 LECTURE NOTES FOR 18.102, SPRING 2009 H1 = {u ∈ H; (u, e1) = 0}. Conclude that S + �B� is invertible as a linear map from H to H1 for small �. Use this to argue that S + �B cannot be an isomorphism from H to H by showing that either e1 is not in the range or else there is a non-trivial element in the null space. Problem 7.4 Show that the product of bounded operators on a Hilbert space is strong continuous, in the sense that if An and Bn are strong convergent sequences of bounded operators on H with limits A and B then the product AnBn is strongly convergent with limit AB. Hint: Be careful! Use the result in class which was deduced from the Uniform Boundedness
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