Inverse of Linear Isometry is Linear Isometry

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

Let $T : X \to Y$ be an invertible (in the sense of a mapping) linear isometry with inverse $T^{-1} : Y \to X$.

Then $T^{-1}$ is a linear isometry.

Proof
From Inverse of Linear Transformation is Linear Transformation, we have:


 * $T^{-1}$ is a linear transformation.

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


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

for each $x \in X$.

Note that for each $y \in Y$, we have $T^{-1} y \in X$.

We then have:

So $T$ is a linear transformation with:


 * $\norm {T^{-1} y}_X = \norm y_Y$

for each $y \in Y$.

So $T^{-1}$ is a linear isometry.