EDB — 2CH

11.2 Isometries[2CH]

We rewrite the definition [0TK]↺↻ in the case of normed spaces.

We will compare it with this definition.

If \(𝜑\) is linear then the definition of equation [(11.24)]↺↻ is equivalent to the definition of linear isometry seen in equation [(11.26)]↺↻ (just set \(z=x-y\)). This explains why both are called ”isometries”.

By the Mazur–Ulam theorem [ if \(M_ 1, M_ 2\) are vector spaces (on real field) equipped with norm and \(𝜑\) is a surjective isometry, then \(𝜑\) is affine (which means that \(x↦ 𝜑(x)-𝜑(0)\) is linear).

We now wonder if there are isometries that are not linear maps, or more generally affine maps.

Exercises

1. [112]↺↻

2. [114]↺↻