Applying Zorn's Lemma (Maximal Subspace)

[Discover85]

Homework Statement

Suppose that V is a vector space, W and X are subspaces with X contained in W. Show that there is a subspace U of V which is maximal subject to the property that U intersect W equals X.

Homework Equations

N/A

The Attempt at a Solution

I know this uses Zorn's Lemma but I can't see how to apply it.

Thanks for any help in advance!

 

[rochfor1]

So what do we need to apply Zorn's lemma? A partially ordered set in which every chain has an upper bound. Let's let our partially ordered set be the family of all subspaces of V whose intersections with W gives X ordered by inclusion. Given a chain in this poset, can you find a natural upper bound?

 

[Discover85]

So what do we need to apply Zorn's lemma? A partially ordered set in which every chain has an upper bound. Let's let our partially ordered set be the family of all subspaces of V whose intersections with W gives X ordered by inclusion. Given a chain in this poset, can you find a natural upper bound?

Would a natural upper bound for this poset be the subspace U such that U contains X and V-{W-X}?

 

[rochfor1]

We're trying to prove that U exists, so we can't use it as the upper bound here. Also, you don't have to find an upper bound for the whole poset, that's what Zorn's lemma gives. You just need to find an upper bound for an arbitrary chain in the poset. A subset S of our poset is a chain if A,B \in S implies that either A\subseteq B or B \subseteq A. What can you say about the union of a family of increasing subpsaces?

 

[Discover85]

Wouldn't the union of increasing subspaces tend to some infinite subspace?

 

[rochfor1]

Take out infinite (which is not necessarily true) and you've got it.

 

[Discover85]

Thanks for your help :)