Definition:Space of Continuous Finite Rank Operators

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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