Intersections of subspaces

Proposition. Let $X$ be a vector space, and let $\{Y_\alpha\}_{\alpha\in I}$ be a family of linear subspaces of $X$ . Then

Y:=\bigcap_{\alpha\in I} Y_\alpha

is also a linear subspace of $X$ .

Remarks

Proof sketch. Each $Y_\alpha$ contains $0$ , so $0\in Y$ . If $x,y\in Y$ , then $x,y\in Y_\alpha$ for every $\alpha$ , hence $x+y\in Y_\alpha$ for every $\alpha$ , so $x+y\in Y$ . Similarly, if $\lambda\in K$ and $x\in Y$ , then $\lambda x\in Y_\alpha$ for all $\alpha$ , so $\lambda x\in Y$ .

This fact underlies the definition of the span as an intersection of all subspaces containing a set.