Completetion of Banach spaces for uncountable index

Question

Let $I$ be an index set, and $V_i$ is a collection of Banach spaces. Consider $p < \infty$, I know that $\sum V_i$ is may not complete under the norm $$||f|| := \left(\sum ||\pi_i(f)||^p\right)^{1/p}$$I know the completion would be functions $f \in \prod V_i$ such that $\sum ||\pi_i(f)||^p < \infty$. This makes complete sense if $I$ is countable since I can just think of $\sum$ as a sequence, but what confuses me is if $I$ is just an index or uncountable(at least what that would look like).