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2. For each of the following, X represents Rx R. Prove that each of the following distance function is a metric on X. Let x = (₁, ₂), y (91, 92) and z = (21, 22) (i).d(x, y) = √(x1 - y₁)² + (x2 - y₂)² (Hint: Use the Minkowski's Inequality: √Σ(an + bn)² ≤ √Σan² + √b² where an, bn are real numbers. (ii). d(x, y) = x₁y₁| + x2 - y2| (iii). d(x, y): = max {x₁y₁, x2 - y2|}.

2. For each of the following, X represents Rx R. Prove that each of the following distance function is a metric on X. Let x = (₁, ₂), y (91, 92) and z = (21, 22) (i).d(x, y) = √(x1 - y₁)² + (x2 - y₂)² (Hint: Use the Minkowski's Inequality: √Σ(an + bn)² ≤ √Σan² + √b² where an, bn are real numbers. (ii). d(x, y) = x₁y₁| + x2 - y2| (iii). d(x, y): = max {x₁y₁, x2 - y2|}.

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. For each of the following, X represents Rx R. Prove that each of the following distance function is a metric on X. Let x = (x₁, x₂), y (9₁, 92) and z = (21, 22) = (i).d(x, y) = √(x₁ - y₁)² + (x2 - y₂)² (Hint: Use the Minkowski's Inequality: /Σ(an + bn)² ≤ √Σan² + √b² where an, bn are real numbers. V (ii). d(x, y) = x₁ - y₁| + x2 - y2| (iii). d(x, y) = max {x₁y₁, x2 - y2|}.
Transcribed Image Text:2. For each of the following, X represents Rx R. Prove that each of the following distance function is a metric on X. Let x = (x₁, x₂), y (9₁, 92) and z = (21, 22) = (i).d(x, y) = √(x₁ - y₁)² + (x2 - y₂)² (Hint: Use the Minkowski's Inequality: /Σ(an + bn)² ≤ √Σan² + √b² where an, bn are real numbers. V (ii). d(x, y) = x₁ - y₁| + x2 - y2| (iii). d(x, y) = max {x₁y₁, x2 - y2|}.
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