Linear isometry

Definition

Suppose $U$ and $V$ are normed vector spaces with the norm

$\|\cdot\|_U$ and

$\|\cdot\|_V$ respectively, a linear isometry preserves norms

$L:U\rightarrow V$ where

$\forall x\in U$ we have

$\|L(x)\|_V=\|x\|_U$

This definition implies

$L$ is injective.

Suppose it were not injective but a linear isometry, then we may have have

$L(a)=L(b)$ and

$a\ne b$ , then

$\|L(a-b)\|_V=\|L(a)-L(b)\|_V=0$ by definition, but as

$a\ne b$ we must have

$\|a-b\|_U>0$ , contradicting that is an isometry.

Thus we can say

$L:U\rightarrow L(U)$ is bijective - but as it may not be onto we cannot say more than

$L$ is injective. Thus

$L$ may not be invertible.

We say that two normed vector spaces are isometric if there is an invertible linear isometry between them.