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Real AnalysisWe started metric spaces yesterday. I am asked to prove that the metric space defined as P<x1,y1> and Q<x2,y2> is a metric space. When I look at examples of these proofs they all say that the proof of the first properties (positive definiteness and symmetry) are trivial. I think I must be more complete than that. How do I best show that these properties are true for this space? Further, how do I show that the triangle inequality is true? Do I create a new ordered pair and call it R<x3,y3> and then prove that the distance between P and Q is less than or equal to the distances from P to R + R to Q?

Question

Asked Nov 23, 2019

Real Analysis

We started metric spaces yesterday. I am asked to prove that the metric space defined as P<x₁,y₁> and Q<x₂,y₂> is a metric space. When I look at examples of these proofs they all say that the proof of the first properties (positive definiteness and symmetry) are trivial. I think I must be more complete than that. How do I best show that these properties are true for this space?

Further, how do I show that the triangle inequality is true? Do I create a new ordered pair and call it R<x₃,y₃> and then prove that the distance between P and Q is less than or equal to the distances from P to R + R to Q?

Expert Answer

Step 1

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The metric properties are, if dis a metric, given as:
Ild must be a real valued, finite and nonnegative.
II d(x, y)0 if and only if x y
d(x, y)d (y,x)
IV] d(x, y)d(x,z)+ d(z,y) — Image Transcription
The metric properties are, if dis a metric, given as: Ild must be a real valued, finite and nonnegative. II d(x, y)0 if and only if x y d(x, y)d (y,x) IV] d(x, y)d(x,z)+ d(z,y)

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