# Linear isometry

## Definition

Suppose [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are normed vector spaces with the norm $\|\cdot\|_U$ and $\|\cdot\|_V$ 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.