Definition:Space of Bounded Linear Transformations

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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:

$\map B {H, K} := \set {A: H \to K: A \text{ linear}, \norm A < \infty}$

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

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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 $H$.

Sources

• 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $\S \text {II}.1$