Does the function d: Y x Y have the same three properties? Explain why or why not.
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]
=
Suppose that the function d
XxX*
Xx X has the following properties: for all f, g, h E X
i. d(f,g) ≥ 0 and d(f, g) = 0 if and only if f = g (i.e. if and
only if f(x) = g(x) for all x = [a, b].);
ii. d(f,g) = d(g, f);
iii. d(f,g) ≤d(f, h) + d(h, g).
Does the function d: Y x Y have the same three properties? Explain why or why not.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2F14939f3f-0bfa-4acf-8c4a-ab863462c4c6%2Fpsixp3e_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]
=
Suppose that the function d
XxX*
Xx X has the following properties: for all f, g, h E X
i. d(f,g) ≥ 0 and d(f, g) = 0 if and only if f = g (i.e. if and
only if f(x) = g(x) for all x = [a, b].);
ii. d(f,g) = d(g, f);
iii. d(f,g) ≤d(f, h) + d(h, g).
Does the function d: Y x Y have the same three properties? Explain why or why not.
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