Uncountable sum of vectors in a Hilbert Space I am currently reading Hilbert Spaces and confused about a thing. Say, $C=\{e_\alpha : \alpha\in\mathcal{A}\}$ be a complete orthonormal set of a Hilbert Space $H$, possibly uncountable. Is $\sum_{\alpha\in\mathcal{A}}e_\alpha$ well defined ? I think it should be, is there some kind of convergence needed for these sums? 

The background of your question is how do you define $\displaystyle \sum_{i \in I} c_i$ in a Banach space for any set $I$?
The usual way is to say that the sum $\displaystyle \sum_{i \in I} c_\alpha$ of a set $\mathcal C = \{c_i \ ; \ i \in I\}$ of vectors exists when following Cauchy criteria is met:
$$(\forall \epsilon > 0) \, (\exists J_0 \in \mathcal F(I)) \, (\forall K \in \mathcal F(I \setminus J_0)) \, \left\Vert \displaystyle \sum_{k \in K} c_k \right\Vert< \epsilon $$
Where $\mathcal F(A)$ is defined as the sets of finite subsets of $A$.
This has interesting consequences.


*

*Such a sum $\displaystyle \sum_{i \in I} c_\alpha$ does make sense only if the set of non zero elements of $\mathcal C$ is at most countable.

*For a family of uncountable vectors of norm equal to $1$, which is your initial question, the sum cannot exist: take $\epsilon = 1/2$ in the definition above; you'll get a contradiction.

