isometric isomorphism

Let $(X,\parallel {\parallel }_X)$ and $(Y,\parallel {\parallel }_Y)$ be normed vector spaces. A surjective linear map $T:X\to Y$ is called an isometric isomorphism between $X$ and $Y$ if

$${\parallel Tx\parallel }_Y={\parallel x\parallel }_X,\text{for all}x\in X.$$

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.