Characterization of Invertible Bounded Linear Transformations

Theorem
Let $\struct {U, \norm \cdot_U}$ and $\struct {V, \norm \cdot_V}$ be normed vector spaces.

Let $A : V \to U$ be a linear transformation with inverse $A^{-1} : U \to V$.

Then $A^{-1}$ is a bounded linear transformation :


 * there exists a real number $c > 0$ such that $\norm {A x}_U \ge c \norm x_V$ for all $x \in V$.

That is, $A$ is invertible as a bounded linear transformation :


 * there exists a real number $c > 0$ such that $\norm {A x}_U \ge c \norm x_V$ for all $x \in V$.

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


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

So we are interested in determining when $A^{-1}$ is bounded.

Necessary Condition
Suppose that $A^{-1}$ was bounded.

Then:


 * there exists a real number $M > 0$ such that $\norm {A^{-1} y}_V \le M \norm y_U$ for all $y \in U$.

Let $x \in V$ and set $y = A x \in U$.

Then:


 * $\norm {A^{-1} y}_V \le M \norm y_U$

We have:

and:


 * $\norm y_U = \norm {A x}_U$

giving:


 * $\norm x_V \le M \norm {A x}_U$

So, we have:


 * $\norm {A x}_U \ge \dfrac 1 M \norm x_V$

for all $x \in V$.

Then, setting $c = 1/M$, we have that:


 * there exists a real number $c > 0$ such that $\norm {A x}_U \ge c \norm x_V$ for all $x \in V$.

Sufficient Condition
Suppose that:


 * there exists a real number $c > 0$ such that $\norm {A x}_U \ge c \norm x_V$ for all $x \in V$.

Let $y \in U$ and let $x = A^{-1} y \in V$.

Then:


 * $\norm {A x}_U \ge c \norm x_V$

We have:

and:


 * $\norm x_V = \norm {A^{-1} y}_V$

so:


 * $\norm y_U \ge c \norm {A^{-1} y}_V$

So, we have:


 * $\norm {A^{-1} y}_V \le \dfrac 1 c \norm y_U$

for all $y \in U$.

So:


 * $A^{-1}$ is bounded.