I was reading some definitions in linear algebra, and I came across the following: Let $V$ be a vector space, and $W_1,W_2$ be its subspaces. The sum of subspaces $W_1+W_2$ is defined as the set $\{w_1+w_2:w_1\in W_1,w_2\in W_2\}$.
Okay, here go my questions:
How does one generalize the definition for $\{W_i\}_{i\in I}$, with $I$ any index set? Is $\sum_{i\in I}W_i$ defined as the set $\{\sum_{i\in I}w_i:w_i\in W_i\}$, or do we impose an additional condition that only finitely many of them can be non-zero?
This applies to not only vector spaces, but also groups, rings, algebras, etc. (and for the sake of argument, I will use groups): If one says that the indexed family of (normal) subgroups $\{N_i\}_{i\in I}$ of $G$ generates $G$, do we assume that each element of $<\bigcup_{i\in I}N_i>$ is a finite product of $n_i$'s for each $n_i\in N_i$?
Thanks!
1) What you wrote at the end: only finite sums are allowed
2) Yes, as before.
In general, for any algebraic structure without any further structure, infinite sums/product/compositions aren't "naturally" defined. For that we would need an additional structure: metric space, inner product and then topology, etc.