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Consider the metric space l∞ whose elements x = {n} are bounded sequences of real numbers, i.e. sup{|xn| : n € N} < ∞. Define a distance d on las below: d(x, y) = sup|xn - Yn. NEN Show that the closed unit ball B centered at 0 = {0} is not compact (although it is bounded and closed).

Consider the metric space l∞ whose elements x = {n} are bounded sequences of real numbers, i.e. sup{|xn| : n € N} < ∞. Define a distance d on las below: d(x, y) = sup|xn - Yn. NEN Show that the closed unit ball B centered at 0 = {0} is not compact (although it is bounded and closed).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1 Consider the metric space l∞ whose elements x = {n} are bounded sequences of real numbers, i.e. sup{|xn| : n € N} < ∞. Define a distance d on lº as below: d(x, y) = sup [xn - Yn|. NEN Show that the closed unit ball B centered at 0= {0} is not compact (although it is bounded and closed).
Transcribed Image Text:1 Consider the metric space l∞ whose elements x = {n} are bounded sequences of real numbers, i.e. sup{|xn| : n € N} < ∞. Define a distance d on lº as below: d(x, y) = sup [xn - Yn|. NEN Show that the closed unit ball B centered at 0= {0} is not compact (although it is bounded and closed).
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