1 Let X be the set of infinite sequences a = (ai)ien of real numbers such...

 

Transcribed Image Text:

1 Let X be the set of infinite sequences a = (ai)ien of real numbers such that º₁|ai|² < ∞. For a = (ai)ieN and b = (bi)ien in X, define and d(a, b): Is da metric? Why? [Justify your assertion with a proof or a counterexample.] 2 Let X be the set of all functions of R into R that are continuous on the unit interval [0, 1]. For f, g E X, define (ai)ieN and b ∞ Σlai - b₁|² i=1 Is d a metric? Why? ||*|| = (S |f(x)|² dx d(f, g) = || f — g||. Is da metric? Why? [Justify your assertion with a proof or a counterexample.] 3 Fix a real number p≥ 1, and let X be the set of infinite sequences a = (ai)ien of real numbers such that a P < 0. For a = (bi)ieN in X, define = ∞ 1/1/20 d(a, b) = |ai - bi|P i=1 1/1/201 «-^/^) ² 1 Let X be the set of infinite sequences a = (ai)ien of real numbers such that º₁|ai|² < ∞. For a = (ai)ieN and b = (bi)ien in X, define and d(a, b): Is da metric? Why? [Justify your assertion with a proof or a counterexample.] 2 Let X be the set of all functions of R into R that are continuous on the unit interval [0, 1]. For f, g E X, define (ai)ieN and b ∞ Σlai - b₁|² i=1 Is d a metric? Why? ||*|| = (S |f(x)|² dx d(f, g) = || f — g||. Is da metric? Why? [Justify your assertion with a proof or a counterexample.] 3 Fix a real number p≥ 1, and let X be the set of infinite sequences a = (ai)ien of real numbers such that a P < 0. For a = (bi)ieN in X, define = ∞ 1/1/20 d(a, b) = |ai - bi|P i=1 1/1/201 «-^/^) ²

Answer rating: 100% (QA)

gons let X be the set of imifinite sequence a an not of real numbers Such that you last ... View the full answer