Let (R>0, d) be the metric space defined by d(x, y) =|log (y/x)|. This metric space is isometric to the Euclidean line E1, where an isometry E1 → (R>0, d) is given by x→ ex . proof that x→ ex is isometric.

Let (R>0, d) be the metric space defined by d(x, y) =|log (y/x)|. This metric space is isometric to the Euclidean line E1, where an isometry E1 → (R>0, d) is given by x→ ex . proof that x→ ex is isometric.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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Question

Let (R_>0, d) be the metric space defined by d(x, y) =|log (y/x)|. This metric space is isometric to the Euclidean line E¹, where an isometry E¹ → (R_>0, d) is given by x→ e^x .

proof that x→ e^xis isometric.

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