Statement: Every vector space has a basis
Standard Proof:It is observed that a maximal linearly independent set is a basis.
Let $\mathscr{Y}$ be a chain of linearly independent subsets of a vector space $\mathscr{V}$. The union of such a set can serve as an upper bound for it.To apply Zorn's lemma,we have to check whether the union is linearly independent? Well, if $t_1,\dots,t_n$ belong to the union, then each $t_i$ belongs to some linearly independent set $L_i\in \mathscr{Y}$. Because $\mathscr{Y}$ is a chain, one of these sets $L_i$ contains all the others. If that is $L_j$, then the linear independence of $L_j$ implies that no non-trivial linear combination of $t_1,\dots,t_n$ can be zero, which proves that the union of the sets in $\mathscr{Y}$ is linearly independent. Therefore, by Zorn’s lemma, there is a maximal linearly independent set and hence a basis.
My question: Why are we using only finite linear combinations to show that the union is linearly independent. Surely, if the union is infinite,then there do exist many infinite linear combinations of elements of the union, which cannot be proven to be linearly independent by the same reasoning. I suspect, that perhaps we are not concerned with infinite linear combinations due to issues of convergence, but I'm not sure.
Clear answers will be appreciated.
Best Answer
The definition of linear independence is that any finite linear relation is trivial.
Vector spaces in general do not have any concept of an infinite sum at all. For those vector space where the usual concept of an infinite sum of reals can be generalized, one may speak of a different kind of span/basis where one allows infinite linear combinations in addition to finite ones. That gives rise to a separate concept, different from the usual kind of linear-combinations basis.
When one needs to distinguish between the different notions of basis, an ordinary basis that works by finite linear combinations is called a "Hamel basis" or "algebraic basis", and one that needs infinite linear combinations to span everything is called a "Schauder basis" (though strictly speaking the latter name implies some additional conditions).