Let be a field. Prove that every -vector space has a basis.
If , then we say is a basis for . Henceforth we assume that is nonzero.
Let be the set of all linearly independent subsets of . Note that is partially ordered by and is nonempty since any nonzero singleton set is linearly independent.
Let be a chain; that is, if , then either or . We claim that is linearly independent. To see this, let be a finite subset and suppose . Now each is in some . Since is finite, there is some index such that . Since is linearly independent, for all . Thus is linearly independent, and thus . That is, the chain has an upper bound in . By Zorn’s lemma, there exists a maximal element .
We claim that is a basis for . To see this, suppose there exists an element . Now let be a finite subset, and suppose . If , then we have , a contradiction. Thus . But since is linearly independent, for all . Thus is linearly independent, violating the maximalness of in . Thus in fact . Thus is a linearly independent generating set (i.e. basis) for .