Results 1 to 2 of 2

Math Help - Union of chains

  1. #1
    Senior Member vincisonfire's Avatar
    Joined
    Oct 2008
    From
    Sainte-Flavie
    Posts
    469
    Thanks
    2
    Awards
    1

    Union of chains

    Hi,
    Suppose that V is a vector space and  \{W_i : i \in I\} is a nonempty collection of subspaces of V.
    I want to show that if  \{W_i : i \in I\} is a chain, then  \cup_{i \in I} W_i  is a subspace of V.
    Intuitively, this is obvious but I don't know how to prove that  \{W_i : i \in I\} is closed under union of chains.
    Then I think I could use Zorn's lemma to say that it has a maximal element that is a subspace of V.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by vincisonfire View Post
    Hi,
    Suppose that V is a vector space and  \{W_i : i \in I\} is a nonempty collection of subspaces of V.
    I want to show that if  \{W_i : i \in I\} is a chain, then  \cup_{i \in I} W_i  is a subspace of V.
    Intuitively, this is obvious but I don't know how to prove that  \{W_i : i \in I\} is closed under union of chains.
    Then I think I could use Zorn's lemma to say that it has a maximal element that is a subspace of V.
    Let W = \cup_{i\in I}W_i. Let \bold{u},\bold{v}\in W then \bold{u}\in W_a and \bold{v} \in W_b for some a,b\in I. However, W_a\subseteq W_b without lose of generality and so \bold{u},\bold{v}\in W_b. This means that \bold{u}+\bold{v} \in W_b\subseteq W. We see that W is closed under vector addition. Argue now that W is closed under scalar multiplication in a similar way. Thus, W is a subspace.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. conclude that the closure of a union is the union of the closures.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 13th 2011, 07:50 PM
  2. Markov Chains
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: February 9th 2010, 05:58 PM
  3. Markov Chains
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: April 26th 2009, 09:17 PM
  4. Markov Chains
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 6th 2009, 10:34 AM
  5. Markov Chains
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 17th 2008, 10:33 AM

Search Tags


/mathhelpforum @mathhelpforum