Prove that infinite intersection of subspaces is a subspace. I was reading a section in a Linear Algebra textbook and I have a small doubt.
I could prove that intersection of any finite collection of subspaces is a subspace. I did this by first doing it for two subspaces, and then extend it easily. But, I am not sure how it would extend to intersection of infinite collection.
My argument for a finite case involves first proving that intersection of two subspaces $W1$ and $W2$ is a subspace. Then let $W12=W1 \cap W2$, and then the same argument will work for $W12$ and $W3$, and so on.
But is the above method enough to conclude that even an intersection of an infinite collection would be a subspace.
Thanks in advance for your help.
 A: You are correct: proving that the intersection of two subspaces is a subspace is enough to conclude that the intersection of finitely many subspaces is a subspace, but not enough to deal with the intersection of infinitely many subspaces.
That said, the proof for the infinite case isn't all too different from the proof in the finite.  Let $\{W_\alpha: \alpha \in I\}$ be a collection of subspaces (for some set $I$ of indices), and let $W = \bigcap_{\alpha \in I}W_{\alpha}$.  Consider any $v_1,v_2 \in W$ and $k \in \Bbb F$. We find that for any $\alpha \in I$, $v_1,v_2 \in W_{\alpha}$, so that $v_1 + kv_2 \in W_{\alpha}$.  By definition, this means that $v_1 + kv_2 \in \bigcap_{\alpha \in I}W_{\alpha} = W$.  
So, $W$ is indeed a subspace.

An example: take $V$ to be the set of all functions $f:\Bbb R \to \Bbb R$.  For any $x \in \Bbb R$, let $V_x = \{f \in V : f(x) = 0\}$.  Let $V_{[0,1]} = \{f \in V : f(x) = 0\text{ for all }x \in [0,1]\}$.
Note that $V_x$ is a subspace for any $x \in \Bbb R$, that $V_{[0,1]} = \bigcap_{x \in [0,1]} V_x$, and that $V_{[0,1]}$ is also a subspace of $V$.
A: The argument is essentially the same. Take two elements in the intersection. Then they belong to all the spaces, hence their sum do so as well. And so on.
