Linear Isometry is Injective

Theorem

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

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

Then $T$ is injective.

Corollary

$T$ is an isometric isomorphism if and only if it is surjective.

Proof

Let $x, y \in X$.

We have:

$\norm {\map T {x - y} }_Y = \norm {T x - T y}_Y$

from the definition of a linear transformation.

Since $T$ is a linear isometry, we have:

$\norm {\map T {x - y} }_Y = \norm {x - y}_X$

So:

$\norm {T x - T y}_Y = 0$

if and only if:

$\norm {x - y}_X = 0$

Since the norm is positive definite, this gives:

$T x = T y$

if and only if:

$x = y$

So $T$ is injective.

$\blacksquare$