6. Let X = C[0, 1] be a metric space with metricd(f, g) := max {|f(x) = g(x)|}.Prove that the sequences {f} given by fn(x) = 2, Vx = [0, 1] is a Cauchy sequences.Which function does it converge to?

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Answered: Let X = C[0, 1] be a metric space with…

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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Let (X, B) be a measurable space and {μn} a sequence ofmeasures with the property that for every E ∈ B,μn(E) ≤ μn+1(E), n = 1, 2,... .Let μ(E) = limn→∞ μn(E). Show that (X, B, μ) is a measurespace.
f, g: A → R be continuous on A, and let h(x) Let = max{ f(x), g(x) } for x belong to A. (a) Show that f (x) + g(x) 1 h(x) +;\f(x) – g(x)| 2 (b) Show that h is continuous on A.
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2. Show that, on the space of continuous functions C[0, 1] the distances defined by the norms = max| |ƒ(x)\, ||f||₁ = √| | \ƒ(1)|dt, ||f||2 = √√√²\ƒ(1)|²³dt, ||f|| = max are mutually non-equivalent.
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Let X = C[0, 1] be a metric space with metric d(f,g) max {f(x) = g(x)}. x[0,1] Prove that the sequences {fn} given by fn(x)=, Vx [0, 1] is a Cauchy sequences. Which function does it converge to?