Difference between revisions of "Linear isometry"

Latest revision as of 11:23, 12 May 2015

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.

@@ Line 1: / Line 1: @@
 ==Definition==
-Suppose {{M|U}} and {{M|V}} are [[Norm|normed]] [[Vector space|vector spaces]] with the norm <math>\|\cdot\|_U</math> and </math>\|\cdot\|_V</math> respectively, a linear isometry preserves [[Norm|norms]]
+Suppose {{M|U}} and {{M|V}} are [[Norm|normed]] [[Vector space|vector spaces]] with the norm <math>\|\cdot\|_U</math> and <math>\|\cdot\|_V</math> respectively, a linear isometry preserves [[Norm|norms]]
 It is a [[Linear map|linear map]] <math>L:U\rightarrow V</math> where <math>\forall x\in U</math> we have <math>\|L(x)\|_V=\|x\|_U</math>
@@ Line 14: / Line 14: @@
 We say that two [[Norm|normed]] [[Vector space|vector spaces]] are isometric if there is an invertible linear isometry between them.
+==Pullback norm==
+*See [[Pullback norm]]