Category:Characterization of Injective Linear Transformations with Closed Image

This category contains pages concerning Characterization of Injective Linear Transformations with Closed Image:

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces.

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

Then $T$ is injective and $\Img T$ is closed if and only if:

there exists $c > 0$ such that $\norm {T x}_Y \ge c \norm x_X$ for each $x \in X$.