Definition:Space of Bounded Linear Transformations

Definition
Let $H, K$ be Hilbert spaces.

Let $\Bbb F \in \set {\R, \C}$ be the ground field of $K$.

Then the space of bounded linear transformations from $H$ to $K$, $\map B {H, K}$, is the set of all bounded linear transformations:


 * $\map B {H, K} := \set {A: H \to K: A \text{ linear}, \text { there exists } M > 0 \text { such that } \norm {A x}_Y \le M \norm x_X \text { for all } x \in X}$

endowed with pointwise addition and ($\Bbb F$)-scalar multiplication.

Then $\map B {H, K}$ is a vector space over $\Bbb F$.

Furthermore, let $\norm {\,\cdot\,}$ denote the norm on bounded linear transformations.

Then $\norm{\,\cdot\,}$ is a norm on $\map B {H, K}$, and it even turns $\map B {H, K}$ into a Banach space.

These results are proved in Space of Bounded Linear Transformations is Banach Space.

Space of Bounded Linear Operators
When $H = K$, one denotes $\map B H$ for $\map B {H, K}$.

In line with the definition of linear operator, $\map B H$ is called the space of bounded linear operators on $H$.