1. ## Union of chains

Hi,
Suppose that V is a vector space and $\displaystyle \{W_i : i \in I\}$ is a nonempty collection of subspaces of V.
I want to show that if $\displaystyle \{W_i : i \in I\}$ is a chain, then $\displaystyle \cup_{i \in I} W_i$ is a subspace of V.
Intuitively, this is obvious but I don't know how to prove that $\displaystyle \{W_i : i \in I\}$ 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.

2. Originally Posted by vincisonfire
Hi,
Suppose that V is a vector space and $\displaystyle \{W_i : i \in I\}$ is a nonempty collection of subspaces of V.
I want to show that if $\displaystyle \{W_i : i \in I\}$ is a chain, then $\displaystyle \cup_{i \in I} W_i$ is a subspace of V.
Intuitively, this is obvious but I don't know how to prove that $\displaystyle \{W_i : i \in I\}$ 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 $\displaystyle W = \cup_{i\in I}W_i$. Let $\displaystyle \bold{u},\bold{v}\in W$ then $\displaystyle \bold{u}\in W_a$ and $\displaystyle \bold{v} \in W_b$ for some $\displaystyle a,b\in I$. However, $\displaystyle W_a\subseteq W_b$ without lose of generality and so $\displaystyle \bold{u},\bold{v}\in W_b$. This means that $\displaystyle \bold{u}+\bold{v} \in W_b\subseteq W$. We see that $\displaystyle W$ is closed under vector addition. Argue now that $\displaystyle W$ is closed under scalar multiplication in a similar way. Thus, $\displaystyle W$ is a subspace.