Cauchy sequence

Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
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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]
Does every uniformly Cauchy sequence in R([a, b]) converge uni-
formly to a function in R([a, b])?
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] Does every uniformly Cauchy sequence in R([a, b]) converge uni- formly to a function in R([a, b])?
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