User:Caliburn/s/fa/Space of Compact Linear Transformations is Linear Subspace of Space of Bounded Linear Transformations

Theorem
Let $\mathbb F$ be a field.

Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\mathbb F$.

Let $\struct {Y, \norm \cdot_Y}$ be a Banach space over $\mathbb F$.

Let $\map {B_0} {X, Y}$ be the space of compact linear transformations $X \to Y$.

Let $\map B {X,Y}$ be the space of bounded linear transformations $X \to Y$.

Then:


 * $\map {B_0} {X, Y}$ is a linear subspace of $\map B {X, Y}$.