Definition:Pseudoinverse of Bounded Linear Transformation

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

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

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

$S$ and $T$ are pseudoinverse to each other :
 * $T \circ S - I_X$ is compact

and:
 * $S \circ T - I_Y$ is compact

where:
 * $\circ$ denotes the composition
 * $I_X$ denotes the identity mapping of $X$
 * $I_Y$ denotes the identity mapping of $Y$