Ordered by inclusion. There is a least element $\{0\}$ and a greatest element $V$. Also for two subspaces $V_1,V_2$ we have $V_1\land V_2 = V_1 \cap V_2$. But what is $V_1\lor V_2$? The union of two subspaces need not be a subspace, right?
/edit: Okay I get that it's a lattice if you take intersection and sum. Now Wikipedia says this lattice is distributive. But what if I have three different 1-dimensional subspaces $A, B, C$ of $\mathbb R^2$, then $$A\cap (B+C) = A \neq \{0\} = (A\cap B)+(A\cap C)$$ Am I wrong?