Lattice of subspaces

Example

Let $V$ be a vector space over a field $F$, and let $L(V)$ be the set of all linear subspaces of $V$, ordered by inclusion. The meet of two subspaces $U$ and $W$ is $U \cap W$, and their join is the sum $U + W = \set{u + w : u \in U, w \in W}$. For any family $(U_i)_{i \in I}$ of subspaces, the intersection $\bigcap_i U_i$ and the span of $\bigcup_i U_i$ are subspaces. They are the greatest lower bound and the least upper bound of the family. So $L(V)$ is a complete lattice, with least element $\set{0}$ and greatest element $V$. It is a modular lattice: if $U \subseteq W$, then $U + (X \cap W) = (U + X) \cap W$ for every subspace $X$. The inclusion $U + (X \cap W) \subseteq (U + X) \cap W$ holds because $U \subseteq W$. For the other inclusion, an element $w = u + x$ of the right side with $u \in U$, $x \in X$ has $x = w - u \in W$, so $x \in X \cap W$.