Weak and norm Borel sigma-algebras in a separable Banach space
= Weak and norm Borel sigma-algebras in a separable Banach space
{title2=$\mathcal B(X,w)=\mathcal B(X,\lVert\cdot\rVert)$}
The <weak topology> and <norm topology> of a <separable Banach space> generate the same <Borel sigma-algebra>. A <countable norming family> makes every norm ball weakly Borel measurable, and separability makes every norm-open set a countable union of such balls. The reverse inclusion follows because the <weak topology> is coarser.