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 it is surjective.

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


 * $T$ is an isometric isomorphism it is bijective.

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


 * $T$ is bijective it is injective and surjective.

From Linear Isometry is Injective, we have:


 * $T$ is injective.

So:


 * $T$ is bijective it is surjective.

Hence:


 * $T$ is an isometric isomorphism it is surjective.