Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D {fE C(X, M); f\х] is bounded } doo(f, 9) := sup d(f(x), 9(x)) xƐX (f,9€ BC(X, М)). Then (BC(X, M), d) is a coтplete metric spасе.
Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D {fE C(X, M); f\х] is bounded } doo(f, 9) := sup d(f(x), 9(x)) xƐX (f,9€ BC(X, М)). Then (BC(X, M), d) is a coтplete metric spасе.
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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![Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and
BC(X, M) :=
ВС (Х,
{ƒ € C(X, M); f[X] is bounded }
:!!
doo(f, 9)
:= sup d(f(x), g(x))
xƐX
(f,9€ BC(Х, М).
Then (BC(X, M), d) is a complete metric space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c566bcb-1b94-430e-9fc2-de3638a71dc9%2F3ece8d9d-6560-4628-bb4f-02e9b940340e%2Fqm2h7u_processed.png&w=3840&q=75)
Transcribed Image Text:Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and
BC(X, M) :=
ВС (Х,
{ƒ € C(X, M); f[X] is bounded }
:!!
doo(f, 9)
:= sup d(f(x), g(x))
xƐX
(f,9€ BC(Х, М).
Then (BC(X, M), d) is a complete metric space.
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