Surjective linear isometry

Let $E$ and $F$ be normed vector spaces. A surjective linear isometry is a surjective linear map $T:E\to F$ satisfying

\lVert Tx\rVert=\lVert x\rVert

for every $x\in E$ .

Consequences

Norm preservation makes $T$ injective, so it is a bijection and its inverse is also a linear isometry.

When $E=F$ , these maps form a group under composition and act on the unit sphere. Two normed spaces are linearly isometric when such a map exists between them.