Let = R([a, b]) be the set (actually a vector space) of bounded, real-valued, Riemann integrable functions on the closed and bounded interval [a, b], and let X = C([a, b]), be the subset (actually a vector subspace) of continuous, real-valued functions on [a, b]. For f, g = Y, define d(f,g) = sup |f(x) — g(x)\. x= [a,b] Show that every uniformly Cauchy sequence in C([a,b]) converges uniformly to a function in C([a,b]).
Let = R([a, b]) be the set (actually a vector space) of bounded, real-valued, Riemann integrable functions on the closed and bounded interval [a, b], and let X = C([a, b]), be the subset (actually a vector subspace) of continuous, real-valued functions on [a, b]. For f, g = Y, define d(f,g) = sup |f(x) — g(x)\. x= [a,b] Show that every uniformly Cauchy sequence in C([a,b]) converges uniformly to a function in C([a,b]).
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 Y= R([a, b]) be the set (actually a vector space) of bounded,
real-valued, Riemann integrable functions on the closed and bounded
interval [a, b], and let X = C([a, b]), be the subset (actually a vector
subspace) of continuous, real-valued functions on [a, b]. For f, g = Y,
define
d(f,g) = sup |f(x) = g(x)\.
x= [a,b]
Show that every uniformly Cauchy sequence in C([a,b]) converges uniformly to a
function in C([a,b]).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2Fad98dcd3-e058-48f2-8792-fbcf03deee1f%2Fym82mep_processed.png&w=3840&q=75)
Transcribed Image Text:Let Y= R([a, b]) be the set (actually a vector space) of bounded,
real-valued, Riemann integrable functions on the closed and bounded
interval [a, b], and let X = C([a, b]), be the subset (actually a vector
subspace) of continuous, real-valued functions on [a, b]. For f, g = Y,
define
d(f,g) = sup |f(x) = g(x)\.
x= [a,b]
Show that every uniformly Cauchy sequence in C([a,b]) converges uniformly to a
function in C([a,b]).
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