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# 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 structure, so they are usually identified with each other.

Defines:

isometrically isomorphic

Related:

Isometry

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

46B99*no label found*

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