Q2 From Q1, we know that du(f.g) = sup{f(r) - g(x): 2 € I} defines a metric on C(I) where I = [0, 1]. (a) Does this still define a metric if the interval I is changed to I = (0, 1)? Explain your answer. (b) Does this still define a metric if I =R? Explain your answer. (e) What if, instead of continuous functions on I= [0, 1], we considered the set of all bounded functions on I? Briefly explain your answer.
Q2 From Q1, we know that du(f.g) = sup{f(r) - g(x): 2 € I} defines a metric on C(I) where I = [0, 1]. (a) Does this still define a metric if the interval I is changed to I = (0, 1)? Explain your answer. (b) Does this still define a metric if I =R? Explain your answer. (e) What if, instead of continuous functions on I= [0, 1], we considered the set of all bounded functions on I? Briefly explain your answer.
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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![Q2 From Q1, we know that du(f.g) = sup{f(r) - g(x): € I} defines a metric on
C(I) where I = [0, 1].
(a) Does this still define a metric if the interval I is changed to I = (0, 1)? Explain
your answer.
(b) Does this still define a metric if I = R? Explain your answer.
(e) What if, instead of continuous functions on I = [0, 1], we considered the set
of all bounded functions on I? Briefly explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55cb5191-da03-417a-8540-4c51a7967251%2Fafe8573f-9f58-4b41-8db8-dd1150d27bf0%2Fau43bch_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q2 From Q1, we know that du(f.g) = sup{f(r) - g(x): € I} defines a metric on
C(I) where I = [0, 1].
(a) Does this still define a metric if the interval I is changed to I = (0, 1)? Explain
your answer.
(b) Does this still define a metric if I = R? Explain your answer.
(e) What if, instead of continuous functions on I = [0, 1], we considered the set
of all bounded functions on I? Briefly explain your answer.
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