Definition:Space of Continuous Finite Rank Operators
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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) $II.4.3$