Linear isometry

Definition

Suppose $U$ and $V$ are normed vector spaces with the norm $$\|\cdot\|_U$$ and </math>\|\cdot\|_V</math> respectively, a linear isometry preserves norms

It is a linear map $$L:U\rightarrow V$$ where $$\forall x\in U$$ we have $$\|L(x)\|_V=\|x\|_U$$

Notes on definition

This definition implies $$L$$ is injective.

Proof

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.

Isometric normed vector spaces

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