Linear Isometry is Injective/Corollary

Corollary

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 an isometric isomorphism if and only if it is surjective.

Proof

From the definition of an isometric isomorphism, we have that:

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

From the definition of a bijective function, we have that:

$T$ is bijective if and only if it is injective and surjective.

From Linear Isometry is Injective, we have:

$T$ is injective.

So:

$T$ is bijective if and only if it is surjective.

Hence:

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

$\blacksquare$