Invertible Continuous Linear Operator has Unique Inverse

Theorem
Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space.

Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.

Let $I \in \map {CL} X$ be the identity element.

Suppose $A \in \map {CL} X$ is invertible.

Then there is a unique $B \in \map {CL} X$ such that $A \circ B = B \circ A = I$.

Proof
Let $B_1, B_2 \in \map {CL} X$.

Suppose:


 * $A \circ B_1 = I = B_1 \circ A$


 * $A \circ B_2 = I = B_2 \circ A$

Then: