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Unformatted text preview: Exercises from Appendix A A.12 Hints: . Let s N = N n =1 f n , and show that the sequence of partial sums { s N } N N is Cauchy in X. . Suppose that every absolutely convergent series is convergent. Let { f n } n N be a Cauchy sequence in X. Show that there exists a subsequence { f n k } k N such that bardbl f n k +1 f n k bardbl &lt; 2 k for every k (see Problem A.2). Then k ( f n k +1 f n k ) is absolutely convergent, hence converges, say to f. Show that { f n } n N has a subsequence that converges (consider the partial sums of k ( f n k +1 f n k )). Show { f n } n N converges (consider Problem A.1). A.17 Hints: There are several ways to prove the Triangle Inequality for 1 &lt; p &lt; . One way is to begin with bardbl x + y bardbl p p = summationdisplay k I  x k + y k  p = summationdisplay k I  x k + y k  p 1  x k + y k  summationdisplay k I  x k + y k  p 1  x k  + summationdisplay k I  x k + y k  p 1  y k  . Then apply H olders Inequality to each sum using the exponent p on the first factor and p for the second (recall that p = p/ ( p 1)). Then divide both sides by bardbl x + y bardbl p 1 p . To show completeness, consider Problem A.6. Show that if { x n } n N is a Cauchy sequence in p and we write x n = ( x n (1) ,x n (2) ,... ) , then for each fixed k we have that { x n ( k ) } n N is a Cauchy sequence of scalars, hence con verges. This gives a candidate sequence x for the limit of { x n } n N . Use the fact that { x n } n N is Cauchy together with the componentwise convergence to show that bardbl x x n bardbl p . A.18 Hint: Assume 0 &lt; p &lt; 1 . Show that (1 + t ) p 1 + t p for t &gt; , and use this to show that ( a + b ) p a p + b p for a , b . A.20 Hints: To show that C c ( R ) is not complete, choose a function in C ( R ) that is nonzero everywhere, and then define appropriate continuous functions g N supported on [ N 1 ,N + 1] such that g N = g on [ N,N ] . Show that { g N } N N is a Cauchy sequence in C c ( R ) that does not converge within C c ( R ) with respect to the uniform norm. On the other hand, this sequence does converge in C ( R ) ....
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 Fall '09
 HEIL
 Topology, Metric space, Open set, accumulation point, Topological space, Closed set

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