Characterization of Finite Rank Operators

Theorem
Let $\HH$ be a Hilbert space.

Let $T \in \map {B_{00} } \HH$ be a bounded finite rank operator.

Let $n = \map \dim {\operatorname {ran} T}$ be the rank of $T$.

Then there are orthonormal vectors $e_1, \ldots, e_n$ and vectors $g_1, \ldots, g_n$ of $\HH$ such that:


 * $\forall h \in \HH: T h = \ds \sum_{i \mathop = 1}^n \innerprod h {e_i}_\HH g_i$