Does d: Y X Y have the same three properties? If so, briefly explain why; if not, identify one property which fails and give an explicit example in which it fails.
Does d: Y X Y have the same three properties? If so, briefly explain why; if not, identify one property which fails and give an explicit example in which it fails.
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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Question
![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 the function d
for all f, g, h € X,
X x X has the following properties:
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);
ii. d(f,g) ≤d(f, h) + d(h, g).
Does d: Y X Y have the same three properties? If so, briefly
explain why; if not, identify one property which fails and give an
explicit example in which it fails.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2F99412cdf-7992-4bfa-83b5-a9b94a9bc93f%2Fi86lhuzc_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 the function d
for all f, g, h € X,
X x X has the following properties:
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);
ii. d(f,g) ≤d(f, h) + d(h, g).
Does d: Y X Y have the same three properties? If so, briefly
explain why; if not, identify one property which fails and give an
explicit example in which it fails.
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