Let $(X, \norm{\ }_X)$ and $(Y, \norm{\ }_Y)$ be normed vector spaces. A surjective linear map $T\colon X \rightarrow Y$ is called an isometric isomorphism between $X$ and $Y$ if $$ \norm{Tx}_Y = \norm{x}_X,\ \mbox{for all}\ x\in X. $$

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.