MATH502_HW2 - Serge Ballif MATH 502 Homework 2 February 1...

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Serge Ballif MATH 502 Homework 2 February 1, 2008 (1) Let a and b be complex numbers with strictly negative real part. Prove the inequality | e a - e b | ≤ | a - b | . Define γ : [0 , 2 π ] to be the straight path γ ( t ) = (1 - t ) a + tb from a to b . Define f ( z ) = e z . Then, since the path γ lies in the left half-plane, we have the inequality | f ( z ) | = e Re z < 1. Thus by the upper bound estimate, we compute Z γ e z d z Length( γ ) . The left-hand side computes to be R 1 0 e γ ( t ) γ 0 ( t ) d t = | e γ ( t ) | 1 0 = | e b - e a | , and the right-hand side computes to be | a - b | , so the result is proved. (2) True or false: There exists a sequence of complex polynomials p n ( z ) such that p n ( z ) 1 /z uniformly on the unit circle { z : | z | = 1 } ? Give careful reasons. Generalizing this, let V be the Banach space of continuous complex-valued functions on the unit circle (with the supremum norm) and let W be the subspace of all functions f for which there exists a sequence of polynomials p n f uniformly, as above. Show that W is a closed subspace of infinite codimen- sion (i.e., dim( V/W ) = ). Seeking a contradiction, we suppose that there exists a sequence of complex poly- nomials p n ( z ) such that p n ( z ) 1 /z uniformly on the unit circle. Let γ be a curve around the unit circle in the counterclockwise direction. Since each poly- nomial is holomorphic, then R γ p n ( z ) d z = 0. We also know that
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