Let $V$ be a vector space, $A$ the collection of all linearly independent subsets of $A$ partially ordered by inclusion $\subset$ and $(C_i)_{i \in I}$ a chain in $A$.
If $\lambda: \bigcup_i C_i \rightarrow \mathbb{F}$ (where $\mathbb{F}$ is a field, e.g. $\mathbb{R}$) is "essentially zero" and $\sum_{c \in \cup_i C_i} \lambda(c) c=0$, then $\lambda(c) \not= 0$ for finitely many $c_{i_1}, ..., c_{i_m}$ in $\bigcup_i C_i$?
Why?
I don't understand why must the finitely many $c_{i_1}, ..., c_{i_m}$ exist, because I thought $\sum_{c \in \cup_i C_i}$ would cover all $c$s that there are and thus there wouldn't be $c$s for which $\lambda(c) \not= 0$.
Or perhaps I misunderstand, what "essentially zero" means.
This appears here, end of p. 1.