Definition:Linear Isometry

Definition

Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.

Let $T : X \to Y$ be a linear transformation.

We say that $T$ is a linear isometry if and only if:

$\norm {T x}_Y = \norm x_X$

for each $x \in X$.

Also see

• Linear Isometry is Injective

• Results about linear isometries can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $3.5$: Isomorphisms between Normed Spaces