Maximal Subspaces in Vector Spaces: Using Zorn's Lemma to Prove Existence

[mathboy]

Maximal subspace

Problem: Prove that every vector space V has maximal subspace, i.e. a proper subspace that is not properly contained in a proper subspace of V.

I let A be the collection of all proper subspaces of V, but I can't prove that every totally ordered subcollection of A has an upper bound in A. The problem that the union of proper subspaces is not necessarily a proper subspace of V. What do I do now?

 

[morphism]

But the union of a chain of subspaces is a subspace.

 

[JasonRox]

Problem: Prove that every vector space V has maximal subspace, i.e. a proper subspace that is not properly contained in a proper subspace of V.

I let A be the collection of all proper subspaces of V, but I can't prove that every totally ordered subcollection of A has an upper bound in A. The problem that the union of proper subspaces is not necessarily a proper subspace of V. What do I do now?

Think basis elements.

 

[andytoh]

But the union of a chain of subspaces is a subspace.

But it has to be a proper subspace of V.

For example { span{1}, span{1,x}, span{1,x,x^2}, span{1,x,x^2,x^3}, ... } is a chain of proper subspaces of R[x], but its union is all of R[x], which is not a proper subspace of R[x].

 

[andytoh]

JasonRox's idea is good, take a basis of V and delete one element. The span of that would have to be a maximal subspace.

But I'm assuming that mathboy wants to use Zorn's lemma. In that case choose any v in V, and let A be the collection of all subspaces not containing v. This time the upper bound of any chain will be a proper subspace. The maximal element of A would be a maximal subspace of V.

 

[morphism]

But it has to be a proper subspace of V.

Oops! I should learn to read! Thanks for pointing that out.

 

[Office_Shredder]

JasonRox's idea is good, take a basis of V and delete one element. The span of that would have to be a maximal subspace.

I don't think the problem implies there IS a basis of V (unless it turns out all vector spaces have a basis, and I just don't know that yet)

 

[JasonRox]

I don't think the problem implies there IS a basis of V (unless it turns out all vector spaces have a basis, and I just don't know that yet)

Is V finite-dimensional? Is the book assuming that?

Do you know what finite-dimensional is?

 

[andytoh]

Every vector space V has a basis, whether it is finite-dimensional or not. In mathboy's problem V can be infinite-dimensional and the result is still true.

If you want to prove that V has a basis if V is infinite-dimensional, you would have to use Zorn's lemma as well. Ultimately, mathboy's problem rests on Zorn's Lemma.


My approach to mathboy's problem is: Choose any v in V, and let A be the collection of all subspaces not containing v and then use Zorn's lemma. But I'm trying to figure out if there is a better partially ordered set to use, because my A seems a little clumsy (though I believe it would still get the job done).

 

[JasonRox]

Every vector space V has a basis, whether it is finite-dimensional or not. In mathboy's problem V can be infinite-dimensional and the result is still true.

Of course I know this!

Ok, a vector space has a basis {v_1,...}, now delete one vector from there and span that that set. What do you get?

Voila!

 

[mathboy]

Thanks guys. I forgot to say that I have to use Zorn's Lemma. But I know how to proceed now. I will use the collection of all proper subspaces that does not contain some fixed v in V.