Denote V 0 by V Observe that a nonempty linearly independent

Denote V {0} by V Observe that a non-empty linearly independent subset of V is an element of P(V ) . Let L P(V ) {} be the set of non-empty linearly independent subsets of V and partially order L by inclusion

(a) Let C be a chain in L. Prove that U= U_{U C} U is an upper bound for C.

(b) Let M be a maximal element of L prove that M spans V .

(c) Now appeal to Zorn’s lemma to prove that V has a basis.

Solution

Suppose a partially ordered set P has the property that every chain has an upper bound in P. Then the set P contains at least one maximal element.