Detailed explanation please! 1Consider the metric space l∞ whose elements x = {n} are bounded sequencesof real numbers, i.e. sup{|xn| : n € N} < ∞. Define a distance d on lº as below:d(x, y) = sup [xn - Yn|.NENShow that the closed unit ball B centered at 0= {0} is not compact (althoughit is bounded and closed).

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Answered: Consider the metric space l∞ whose…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 78E

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Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f E C(X, M); f[X] is bounded } doo(f, 9) := sup d(f(x), g(x)) xƐX (f, g € BC(X, M)). Then (BC(X, M), d) is a complete metric space.

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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).