isometric isomorphism

Let $(X,\left\|\ \right\|_{X})$ and $(Y,\left\|\ \right\|_{Y})$ be normed vector spaces. A surjective linear map $T\colon X\rightarrow Y$ is called an isometric isomorphism between $X$ and $Y$ if

 $\left\|Tx\right\|_{Y}=\left\|x\right\|_{X},\ \mbox{for all}\ x\in X.$

In this case, $X$ and $Y$ are said to be isometrically isomorphic.

Two isometrically isomorphic normed vector spaces share the same , so they are usually identified with each other.

Title isometric isomorphism IsometricIsomorphism 2013-03-22 17:34:17 2013-03-22 17:34:17 Gorkem (3644) Gorkem (3644) 8 Gorkem (3644) Definition msc 46B99 Isometry isometrically isomorphic