Let $M$ and $N$ be von Neumann algebras, and let $\Phi:M\to N$ be a positive [linear map](/page/Linear%20Map). Then $\Phi$ is normal if and only if, for every bounded subset $B\subset M$, the restriction $\Phi|_B:B\to N$ is continuous from the relative ultraweak topology on $B$ to the ultraweak topology on $N$. Equivalently, for every bounded net $(x_i)_{i\in I}$ in $M$ and every $x\in M$, if $x_i\to x$ ultraweakly in $M$, then $\Phi(x_i)\to \Phi(x)$ ultraweakly in $N$.