 Show that if /:X— > Y is uniformly continuous and {xn} is a Cauchy sequence in X, then {/(*„)} is a Cauchy sequence in Y.  Real Analysis - Seite 385von Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson - 1997 - 713 SeitenVollansicht - Über dieses Buch
 | Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner - 2008 - 685 Seiten
...ye A}. In Exercise 13.6.1 1 we established that / is continuous. Is / uniformly continuous? 13.12.16 Show that if /:X— > Y is uniformly continuous and {xn} is a Cauchy sequence in X, then {/(*„)} is a Cauchy sequence in Y. Show that this need not be true if / is merely continuous. 13.12.17... | |
 | Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner - 2001 - 753 Seiten
...ye A}. In Exercise 13.6.11 we established that / is continuous. Is / uniformly continuous? 13.12.16 Show that if / : X — > Y is uniformly continuous and {xn} is a Cauchy sequence in X, then {f(xn)} is a Cauchy sequence in Y. Show that this need not be true if / is merely continuous. 13.12.17... | |
 | George Bachman, Lawrence Narici - 2000 - 548 Seiten
...continuous linear functional defined on the normed linear space X. Prove that, if the sequence {*N} is a Cauchy sequence in X, then {/(xn)} is a Cauchy sequence of complex numbers. 7. If p is a convex functional on X, show that, for arbitrary x0 e X and a > 0,... | |
 | Carlos S. Kubrusly - 2001 - 546 Seiten
...3.8 and Lemma 3.43). // (xn\ converges in X to xe X , then {[xn]} converges in X/M to [x] e X/M', if [xn] is a Cauchy sequence in X, then {[xn]\ is a Cauchy sequence in X/M. Proposition 4.10. If M is a subspace of a Banach space X, then the quotient space X/M is a Banach... | |
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