Weak and norm Borel sigma-algebras in a separable Banach space ($\mathcal{B}(X,w)=\mathcal{B}(X,\Vert \cdot \Vert )$)

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.