# Local base

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• Feb 24th 2009, 05:18 PM
dori1123
Local base
Let $B$ be a local base for a topological vector space $X$. Let $U$ be a neighborhood of 0 in $X$. Show that $\cap_{W \in B} (U \cap W) = \cap_{W \in B} W$.
• Feb 26th 2009, 08:26 AM
tah
Obviously, $\cap_{W\in B}(U\cap W)\subseteq\cap_{W\in B}W$.
Let $x\in \cap_{W\in B}W$ and $O\in B$ such that $O\subseteq U$ (definition) then $x\in O\subseteq U$ i.e. $x\in U\cap W$ for any $W\in B$