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2.1. If X and Y are two metric spaces, then a function f : X ? Y is an isometry means that it is a bijection that preserves distances: d(p,q) = d(f(p), f(q)). (a) Let X = ?² be the plane with the taxicab norm ||?v||? and corresponding metric d?(?v, ?w), and let Y = ?² be the plane with the sup norm ||?v||? and the corresponding metric d?(?v, ?w). Prove that there is an isometry between X and Y. (Hint: There is one that is a linear transformation.) (b) Let X = ?² be the plane with the taxicab norm ||?v||? and corresponding metric d?(?v, ?w), and let Y = ?² be the plane with the Euclidean norm ||?v||? and the corresponding metric d?(?v, ?w). Prove that there isn't isometry between X and Y. (Hint: Isometric paths are always unique in one of X and Y, but not in the other one.) *(c) Prove that there is no linear isometry between ?³ with the taxicab norm ||?v||? and ?³ with the sup norm ||?v||?

          2.1. If X and Y are two metric spaces, then a function f : X ? Y is an isometry means that it is a bijection that preserves distances: d(p,q) = d(f(p), f(q)).

(a) Let X = ?² be the plane with the taxicab norm ||?v||? and corresponding metric d?(?v, ?w), and let Y = ?² be the plane with the sup norm ||?v||? and the corresponding metric d?(?v, ?w). Prove that there is an isometry between X and Y. (Hint: There is one that is a linear transformation.)

(b) Let X = ?² be the plane with the taxicab norm ||?v||? and corresponding metric d?(?v, ?w), and let Y = ?² be the plane with the Euclidean norm ||?v||? and the corresponding metric d?(?v, ?w). Prove that there isn't isometry between X and Y. (Hint: Isometric paths are always unique in one of X and Y, but not in the other one.)

*(c) Prove that there is no linear isometry between ?³ with the taxicab norm ||?v||? and ?³ with the sup norm ||?v||?

2.1. If X and Y are two metric spaces, then a function f : X ? Y is an isometry means that it is a bijection that preserves distances: d(p,q) = d(f(p), f(q)).

(a) Let X = ?² be the plane with the taxicab norm ||?v||? and corresponding metric d?(?v, ?w), and let Y = ?² be the plane with the sup norm ||?v||? and the corresponding metric d?(?v, ?w). Prove that there is an isometry between X and Y. (Hint: There is one that is a linear transformation.)

(b) Let X = ?² be the plane with the taxicab norm ||?v||? and corresponding metric d?(?v, ?w), and let Y = ?² be the plane with the Euclidean norm ||?v||? and the corresponding metric d?(?v, ?w). Prove that there isn't isometry between X and Y. (Hint: Isometric paths are always unique in one of X and Y, but not in the other one.)

*(c) Prove that there is no linear isometry between ?³ with the taxicab norm ||?v||? and ?³ with the sup norm ||?v||?

Added by Juana J.

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