Let $V$ be a vector space over a division ring $D$ and $S$ the set of all subspaces on $V$, partially ordered by set theoretic inclusion.
(i) $S$ is a complete lattice
(ii) $S$ is a complemented lattice; that is for each $V_1\in S$ there exists $V_2\in S$ such that $V=V_1+V_2$ and $V_1\bigcap V_2=0$, so that $V=V_1\oplus V_2$.
(iii) $S$ is a modular lattice; that is, if $V_1,V_2,V_3\in S$ and $V_3\subset V_1$ then $$V_1\cap (V_2+V_3)=(V_1\cap V_2)+V_3.$$
I need help with this problem. I'm not sure where to start. I think I understand what a complete lattice, a complemented lattice and a modular lattice. I'm just not sure how to start. I have ideas but I'm not sure if they are correct. For example I think the LUB subspace is the intersection of all subspaces while the GLB is the vector space V. Any help/hint would be appreciated. I hope somebody could point me in the right direction. Thanks!