Exercise 2. Consider F(X, Y) the space of all maps f: X→ Y from the set (Y, d). For f,g: X→Y, define do(f, g) = [0, +∞] by do(f, g) = sup d(f(x), g(x)) = [0, +∞]. TEX 1) Show the following properties of doo: (i) do(f, g) = 0 if and only if f g everywhere on X; (ii) (Symmetry) do(f,g) = doo (9, f); (iii) (Triangular Inequality) do(f, h) ≤ do(f, g) + do(g, h). to the metric space
Exercise 2. Consider F(X, Y) the space of all maps f: X→ Y from the set (Y, d). For f,g: X→Y, define do(f, g) = [0, +∞] by do(f, g) = sup d(f(x), g(x)) = [0, +∞]. TEX 1) Show the following properties of doo: (i) do(f, g) = 0 if and only if f g everywhere on X; (ii) (Symmetry) do(f,g) = doo (9, f); (iii) (Triangular Inequality) do(f, h) ≤ do(f, g) + do(g, h). to the metric space
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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Exercise 2
Need each part I, II, III
![Exercise 2. Consider F(X,Y) the space of all maps f: X→Y from the set X to the metric space
(Y, d). For f,g: X →Y, define do(f,g) = [0, +∞] by
do(f, g) = sup d(f(x), g(x)) = [0, +∞].
xEX
1) Show the following properties of do:
(i) do(f, g) = 0 if and only if f = g everywhere on X;
(ii) (Symmetry) dx (f, g) = do(g, f);
(iii) (Triangular Inequality) do(f, h) ≤ do(f, g) + do(g, h).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2F47b50f94-127a-4ad4-8569-971483e299c4%2Fuj21sjl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 2. Consider F(X,Y) the space of all maps f: X→Y from the set X to the metric space
(Y, d). For f,g: X →Y, define do(f,g) = [0, +∞] by
do(f, g) = sup d(f(x), g(x)) = [0, +∞].
xEX
1) Show the following properties of do:
(i) do(f, g) = 0 if and only if f = g everywhere on X;
(ii) (Symmetry) dx (f, g) = do(g, f);
(iii) (Triangular Inequality) do(f, h) ≤ do(f, g) + do(g, h).
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