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2. Answer the following questions about metric spaces. All the theories we are discussing in this course are essentially properties of metric spaces. i) What is the definition of Metric Space (M, d)? Here, M is the set and d is the metric defined on M. ii) Show that (R³, d₁(, )) is metric space by checking the definition of metric space. Here we define when xy d₁(x, y) 0 when x = Y. iii) Let F be the collection of all continuous (real-valued) functions on the unit interval [0, 1]. If we define d₂(fi; f2) = [" \f₁(x) — ƒ2(x)| da. 0 Show that (F, d₂) is a metric space. (Don't forget to show that d2(f1, f2) = 0 only when f₁ = f2.)

2. Answer the following questions about metric spaces. All the theories we are discussing in this course are essentially properties of metric spaces. i) What is the definition of Metric Space (M, d)? Here, M is the set and d is the metric defined on M. ii) Show that (R³, d₁(, )) is metric space by checking the definition of metric space. Here we define when xy d₁(x, y) 0 when x = Y. iii) Let F be the collection of all continuous (real-valued) functions on the unit interval [0, 1]. If we define d₂(fi; f2) = [" \f₁(x) — ƒ2(x)| da. 0 Show that (F, d₂) is a metric space. (Don't forget to show that d2(f1, f2) = 0 only when f₁ = f2.)

Advanced Engineering Mathematics
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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2. Answer the following questions about metric spaces. All the theories we are discussing in this course are essentially properties of metric spaces. i) What is the definition of Metric Space (M, d)? Here, M is the set and d is the metric defined on M. ii) Show that (R³, d₁(, )) is metric space by checking the definition of metric space. Here we define when xy when x = y. iii) Let F be the collection of all continuous (real-valued) functions on the unit interval [0, 1]. If we define - d₁(x, y) · S'* \f₁(x) — £2(x)\ dæ. d₂(f₁, f2) = √ ² 1/fi Show that (F, d₂) is a metric space. (Don't forget to show that d2(f1, f2) = 0 only when f₁ = f2.)
Transcribed Image Text:2. Answer the following questions about metric spaces. All the theories we are discussing in this course are essentially properties of metric spaces. i) What is the definition of Metric Space (M, d)? Here, M is the set and d is the metric defined on M. ii) Show that (R³, d₁(, )) is metric space by checking the definition of metric space. Here we define when xy when x = y. iii) Let F be the collection of all continuous (real-valued) functions on the unit interval [0, 1]. If we define - d₁(x, y) · S'* \f₁(x) — £2(x)\ dæ. d₂(f₁, f2) = √ ² 1/fi Show that (F, d₂) is a metric space. (Don't forget to show that d2(f1, f2) = 0 only when f₁ = f2.)
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