What is the Proper Subspace Hierarchy in a Vector Space with Infinite Subsets?

[mathboy]

Homework Statement

Let {W_1,W_2,W_3,...} be a collection of proper subspaces of V (i.e. W_i not=V) such that W_i is a subset of W_(i+1) for all i. Prove that U(W_i) (i from 1 to infinity) is a proper subspace of V

The Attempt at a Solution

I've already proven that U(W_i) is a subspace of V, so I only need to show that U(W_i) not= V. I've used induction but that only proves that W_i (i from 1 to n) is a proper subset of V, not U(W_i) (i from 1 to infinity). How do I show that U(W_i) (i from 1 to infinity) is a proper subset of V? I'm familiar with infinite set theory stuff (axiom of choice, etc...), but I don't know how to use it here. I could not use dimensions to help me because infinity minus a number is still infinity. Should I use the complement subsets of W_i?

 

[Dick]

Why can't you DEFINE V to be the union of the W's. Doesn't that V meet all of your premises? But the union of the W's now IS V. I don't see how you could prove the union is a proper subspace.

 

[mathboy]

V is given. I cannot give it any definition. {W_1,W_2,W_3,...} is defined to be a collection of proper subspaces of V. I can define Y to be the union of the W's, but then I have to show that Y is a proper subset of V.

 

[Dick]

I'm pointing out that Y is a counterexample to what you are trying to prove. So what you are trying to prove can't be true for all V.

 

[Dick]

Or maybe there is something you haven't told us about V. Does it have some sort of completeness property?

 

[HallsofIvy]

For a specific counterexample, let V be the vector space of all polynomials. For every n, let W_n be the subspace of all polynomials of degree less than or equal to n. I believe that satisfies the conditions. The union of all such spaces is V itself.