It seems to be a well-known fact [1] that basis vectors $|e_i\rangle$ of finite-dimensional Hilbert space $\mathbb{H}$ and the vectors of corresponding dual basis $\langle e_j| \in \mathbb{H} \to_{lin} \mathbb{C}$ could be combined in diads $|e_i\rangle\langle e_j| \in \mathbb{H}\to_{lin}\mathbb{H}$ such that $\forall |v\rangle\in\mathbb{H} : (|e_i\rangle\langle e_j|) |v\rangle = \langle e_j|v\rangle|e_i\rangle$ where $\langle .|.\rangle \in \mathbb{C}$ denotes the inner product in $\mathbb{H}$.
How to proof that such $|e_i\rangle\langle e_j|$ actually form a basis in the space of linear functions $\mathbb{H}\to_{lin}\mathbb{H}$, that is $\forall f\in \mathbb{H}\to_{lin}\mathbb{H}: \exists F_{ij}\in\mathbb{C}: f = \sum_{ij}F_{ij}|e_i\rangle\langle e_j|$ ?
[1] - Vannucci, Tensor Algebra and Tensor Analysis, page 8
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Updates: Added a note about finite dimensionality of $\mathbb{H}$.