Question

A mapping f : X ? Y between metric spaces (X,d) and (Y,?) is called an isometry if it preserves distances, i.e., ?(f(a), f(b)) = d(a,b) for all a, b ? X. (a) Show that an isometry must be injective (one to one). (b) Is an isometry necessarily surjective (onto)? Prove or provide a counterexample. (c) Is there a metric space which is isometric to a proper subset of itself? Provide either a proof that such a space cannot exist, or an example of such a space. (d) Is there a normed space which is isometric to a proper subspace of itself? Provide either a proof that such a space cannot exist, or an example of such a space. (e) Is ? with its usual metric isometric to a proper subset of itself? Prove your answer.

          A mapping f : X ? Y between metric spaces (X,d) and (Y,?) is called an isometry if it preserves distances, i.e., ?(f(a), f(b)) = d(a,b) for all a, b ? X.

(a) Show that an isometry must be injective (one to one).

(b) Is an isometry necessarily surjective (onto)? Prove or provide a counterexample.

(c) Is there a metric space which is isometric to a proper subset of itself? Provide either a proof that such a space cannot exist, or an example of such a space.

(d) Is there a normed space which is isometric to a proper subspace of itself? Provide either a proof that such a space cannot exist, or an example of such a space.

(e) Is ? with its usual metric isometric to a proper subset of itself? Prove your answer.

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A mapping f : X ? Y between metric spaces (X,d) and (Y,?) is called an isometry if it preserves distances, i.e., ?(f(a), f(b)) = d(a,b) for all a, b ? X.

(a) Show that an isometry must be injective (one to one).

(b) Is an isometry necessarily surjective (onto)? Prove or provide a counterexample.

(c) Is there a metric space which is isometric to a proper subset of itself? Provide either a proof that such a space cannot exist, or an example of such a space.

(d) Is there a normed space which is isometric to a proper subspace of itself? Provide either a proof that such a space cannot exist, or an example of such a space.

(e) Is ? with its usual metric isometric to a proper subset of itself? Prove your answer.

Added by Bego-A T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/18/2023

Step 1

Suppose $f(a) = f(b)$ for some $a, b \in X$. Then, by the definition of an isometry, we have $p(f(a), f(b)) = d(a, b)$. Since $f(a) = f(b)$, we have $p(f(a), f(a)) = d(a, b)$. But $p(f(a), f(a)) = 0$ because the distance between a point and itself is always 0. Show more…

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