# Every vector space has a basis using minimal spanning set.

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?

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It doesn't work. See Keith Conrad's note (namely page 16). Here is a relevant screenshot.

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Ah, good answer, and good question! I misread the intent of the original question. –  rschwieb Apr 22 at 13:15
@rschwieb: Thanks! :) I liked your (now-deleted) answer too. –  Prism Apr 22 at 13:20
Actually, as I keep rereading and figuring out what the OP actually meant, my answer needs to be edited. –  rschwieb Apr 22 at 13:22

Does the notion of minimal spanning set make sense for arbitrary vector spaces?

Sure: why not? The notion has a sensible definition. A minimal spanning set $S$ is one for which $\langle S'\rangle \subsetneq \langle S\rangle$ whenever $S'$ is a proper subset of $S$.

If instead $S'$ were a proper subset of $S$, and yet $\langle S'\rangle = \langle S\rangle$, then it would follow that every element of $S\setminus S'$ is a linear combination of the elements of $S'$, and hence linearly dependent upon the elements of $S'$. So, elements could be removed from $S$ while preserving the span, and $S$ would not be a minimal spanning set.

..can the statement that every vector space has a basis be proved using the partially ordered set $\Sigma = \lbrace A \subset V \vert Span(A) =V \rbrace$ and define partial order $A \leq B$ iff $B \subset A$. Can one say intersection of all totally ordered spanning set is also a spanning set?