Characterization of Finite Rank Operators
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Theorem
Let $H$ be a Hilbert space.
Let $T \in B_{00} \left({H}\right)$ be a bounded finite rank operator.
Let $n = \operatorname{dim} \left({\operatorname{ran} T}\right)$ be the rank of $T$.
Then there are orthonormal vectors $e_1, \ldots, e_n$ and vectors $g_1, \ldots, g_n$ of $H$ such that:
- $\forall h \in H: Th = \displaystyle \sum_{i=1}^n \left\langle{h, e_i}\right\rangle_H g_i$
Proof
Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $II.4 \text{ Exercise } 8$