# Characterization of Finite Rank Operators

## 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:

$\forall h \in H: Th = \displaystyle \sum_{i=1}^n \left\langle{h, e_i}\right\rangle_H g_i$