Definition:Space of Bounded Linear Transformations

Definition
Let $H, K$ be Hilbert spaces.

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

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


 * $B \left({H, K}\right) := \left\{{A: H \to K: A \text{ linear}, \left\Vert{A}\right\Vert < \infty}\right\}$

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

Then $B \left({H, K}\right)$ is a vector space over $\Bbb F$, as proved in Space of Bounded Linear Transformations is Vector Space.

Furthermore, let $\left\Vert{\cdot}\right\Vert$ denote the norm on linear transformations.

Then $\left\Vert{\cdot}\right\Vert$ is a norm on $B \left({H, K}\right)$, as proved in Space of Bounded Linear Transformations is Normed Vector Space.

That is, $B \left({H, K}\right)$ is a normed vector space.

Space of Bounded Operators
When $H = K$, one denotes $B \left({H}\right)$ for $B \left({H, K}\right)$.

In line with the definition of operator, $B \left({H}\right)$ is called the space of bounded operators on $H$.