We have seen the argument for proving the above statment using Zorn's Lemma by asserting the existence of a maximal linearly independent set which serves a basis. In finite dimensional vector space, a basis is same as a maximal linearly independent and also same as a minimal spanning set.
Does the notion of minimal spanning set make sense for arbitrary vector spaces?
Moreover, can the statement that every vector space has a basis be proved using the partially ordered set $\Sigma = \lbrace A \subset V \vert Span(A) =V \rbrace $ with the partial order $A \leq B$ iff $B \subset A $?
Can one say intersection of a chain of spanning sets in this poset is also a spanning set?
Best Answer
It doesn't work. See Keith Conrad's note (namely page 16). Here is a relevant screenshot.