Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M); f[X] is bounded } d∞(f, g) := sup d(f(x), g(x)) (f, g ∈ BC(X, M)). Then (BC(X, M), d∞) is a complete metric space.

Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M); f[X] is bounded } d∞(f, g) := sup d(f(x), g(x)) (f, g ∈ BC(X, M)). Then (BC(X, M), d∞) is a complete metric space.

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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Question

Let (X, T ) be a topological space, (M, d) be a complete metric space and
BC(X, M) := {f ∈ C(X, M); f[X] is bounded }
d∞(f, g) := sup d(f(x), g(x)) (f, g ∈ BC(X, M)).
Then (BC(X, M), d∞) is a complete metric space.

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