Let X be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X Show that d(f, g) = sup{|f(x) – g(x)| : x € X} defines a metric on C(X), so that C(X) becomes a complete metric space.
Let X be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X Show that d(f, g) = sup{|f(x) – g(x)| : x € X} defines a metric on C(X), so that C(X) becomes 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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![Let \( X \) be a compact subset of \( \mathbb{R} \) and let \( C(X) \) denote the set of continuous real-valued functions on \( X \).
Show that
\[ d(f, g) = \sup \{ |f(x) - g(x)| : x \in X \} \]
defines a metric on \( C(X) \), so that \( C(X) \) becomes a complete metric space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F916f3b4b-5515-4e49-adba-db29d1d6eb22%2Fdb91520d-63be-412d-b2b6-d2ba010416b2%2Fmymywek_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( X \) be a compact subset of \( \mathbb{R} \) and let \( C(X) \) denote the set of continuous real-valued functions on \( X \).
Show that
\[ d(f, g) = \sup \{ |f(x) - g(x)| : x \in X \} \]
defines a metric on \( C(X) \), so that \( C(X) \) becomes a complete metric space.
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