Image of Weakly Convergent Sequence under Compact Linear Transformation is Convergent

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

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

Let $\sequence {x_n}_{n \mathop \in \N}$ be a weakly convergent sequence with:


 * $x_n \weakconv x$

Then:


 * $T x_n \to T x$

in the strong sense.