Let $\displaystyle B$ be a local base for a topological vector space $\displaystyle X$. Let $\displaystyle U$ be a neighborhood of 0 in $\displaystyle X$. Show that $\displaystyle \cap_{W \in B} (U \cap W) = \cap_{W \in B} W$.
Obviously, $\displaystyle \cap_{W\in B}(U\cap W)\subseteq\cap_{W\in B}W$.
Let $\displaystyle x\in \cap_{W\in B}W$ and $\displaystyle O\in B$ such that $\displaystyle O\subseteq U$ (definition) then $\displaystyle x\in O\subseteq U$ i.e. $\displaystyle x\in U\cap W$ for any $\displaystyle W\in B$