Given a Hilbert space $\cal H$, what criterion describes the property "$\cal B$ is a Hilbert basis for $\cal H$"? It would be even better if the definition can be stated in a way that mimics some characterization of a vector space basis.
For example, if we consider the spanning set part of the definition, we have
- A vector space basis satisfies ${\rm span}({\cal B})=V$,
which has the analogue
- A Hilbert space basis satisfies $\overline{{\rm span}({\cal B})}=\cal H$.
What are the analogues of vector sum characterizations? Or linear independence?
In a Hilbert space, one typically uses a orthonormal basis, that is a system of vectors $\{x_n\}$, with $$(x_i, x_j) = \delta_{ij}$$ and $$\overline{\mathrm{span}(\{x_n\}) } = \mathcal{H}.$$