User:Caliburn/s/fa/Bijective Bounded Linear Transformation is Linear Isometry iff Transformation and Inverse have Norm 1

Theorem

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

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

Let $T^{-1} : Y \to X$ be the inverse linear transformation of $T$.

Then:

$T$ is a linear isometry

if and only if:

$T^{-1}$ is bounded

and:

$\norm T = \norm {T^{-1} } = 1$

where $\norm \cdot$ is the norm of a bounded linear transformation.