# Definition:Space of Continuous Finite Rank Operators

## Definition

Let $H, K$ be Hilbert spaces.

Then the **space of continuous finite rank operators from $H$ to $K$**, denoted $B_{00} \left({H, K}\right)$, is the set:

- $B_{00} \left({H, K}\right) := \left\{{A \in B \left({H, K}\right): A \text{ is of finite rank} }\right\}$

of all bounded linear transformations of finite rank.

By definition, it is a subset of the space of bounded linear transformations $B \left({H, K}\right)$.

In fact, by Finite Rank Operator is Compact, it is contained in $B_0 \left({H, K}\right)$, the space of compact linear transformations.

## Also see

- Finite Rank Operator
- Space of Bounded Linear Transformations
- Space of Compact Linear Transformations
- Finite Rank Operators Dense in Compact Linear Transformations

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.4.3$