Exercise 1067Topology › Metric and normed spacesL1Added 3 Nov 2026

Isometries between metric spaces

#isometry#metric#distance-preserving

Statement

Define an isometry and show it is injective.

Hint

If $f(x)=f(y)$ , apply $d(f(x),f(y))=d(x,y)$ .

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Definition. $f:X\to Y$ is an isometry if $d_Y(f(x),f(y))=d_X(x,y)$ for all $x,y$ .

Injectivity. $f(x)=f(y)\Rightarrow d_Y(f(x),f(y))=0\Rightarrow d_X(x,y)=0\Rightarrow x=y$ .

Conclusion. Every isometry is injective; if surjective, it is an isometric isomorphism.