3.1 Let S be the set of all bounded continuous functions f: [a,b] → R. Given two points f and g in X, let d(f.g) = sup{\f(x) - g(x) = x= [a,b]} Determine whether d is a metric on S or not. 3.2 Define a bounded subset of a metric space (S,d). 3.3 Show that a union of any finite number of bounded subsets of a metric space is bounded.
3.1 Let S be the set of all bounded continuous functions f: [a,b] → R. Given two points f and g in X, let d(f.g) = sup{\f(x) - g(x) = x= [a,b]} Determine whether d is a metric on S or not. 3.2 Define a bounded subset of a metric space (S,d). 3.3 Show that a union of any finite number of bounded subsets of a metric space is bounded.
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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![3.1 Let S be the set of all bounded continuous functions f: [a,b] → R. Given two points f
and g in X, let
d(f,g) = sup{f(x) = g(x) = x= [a, b]}
Determine whether d is a metric on S or not.
3.2 Define a bounded subset of a metric space (S, d).
3.3 Show that a union of any finite number of bounded subsets of a metric space is
bounded.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5eb62559-4f34-4af6-9980-4ef033b0a109%2F815dfca5-2d83-4916-bc48-dc85471cc406%2F1glfz2a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.1 Let S be the set of all bounded continuous functions f: [a,b] → R. Given two points f
and g in X, let
d(f,g) = sup{f(x) = g(x) = x= [a, b]}
Determine whether d is a metric on S or not.
3.2 Define a bounded subset of a metric space (S, d).
3.3 Show that a union of any finite number of bounded subsets of a metric space is
bounded.
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