QuestionQuestion

1. Suppose (X,d1) is a metric space and define d­2: X × X àℝ by d­2= d1 (x,y)/1 + d1 (x,y). Is (X,d­2) a metric space? Prove your claim.

2. Definition: Suppose (X,d1) and (X,d­2) are metric spaces. We say d1 equivalent to d­2 if there exists positive constants m and M such that md­2(x,y) ≤ d1(x,y) ≤ Md­2(x,y) for all x and y in X.

Define
d1 :ℝ­­2 × ℝ­­2 by d1 ((x1,y1), (x2, y2)) = √ (x2 – x1)2 + (y2 – y1)2
d2 : ℝ­­2 × ℝ­­2 by d2 ((x1,y1), (x2, y2)) = max {lx2 – x1l, ly2 – y1l}
d3 : ℝ­­2 × ℝ­­2 by d3 ((x1,y1), (x2, y2)) = lx2 – x1l + ly2 – y1l
(a) Show d1is equivalent to d­2
(b) Show d3 is equivalent to d1

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