Definition:Space of Compact Linear Transformations
Definition
Let $H, K$ be Hilbert spaces.
Let $\Bbb F \in \left\{{\R, \C}\right\}$ be the ground field of $K$.
The space of compact linear transformations from $H$ to $K$, $B_0 \left({H, K}\right)$, is the set of all compact linear transformations:
- $B_0 \left({H, K}\right):= \left\{{T: H \to K: T \text{ compact}}\right\}$
endowed with pointwise addition and ($\F$)-scalar multiplication.
It is a Banach space, as proven on Space of Compact Linear Transformations is Banach Space.
The notation resembles that for the space of bounded linear transformations $B \left({H, K}\right)$.
This is appropriate as a Compact Linear Transformation is Bounded; i.e., $B_0 \left({H, K}\right) \subseteq B \left({H, K}\right)$.
Space of Compact Linear Operators
When $H$ is equal to $K$, one speaks about the space of compact (linear) operators instead.
One writes $B_0 \left({H}\right)$ for $B_0 \left({H, H}\right)$.
Also see
- Space of Compact Linear Transformations is Banach Space
- Compact Linear Transformation
- Space of Bounded Linear Transformations
Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $II.4.1$