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

is also a linear subspace of $X$.

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.