Suppose $T\_{n}$ converging $T$ in vN algebra $M$ in weak operator topology, can we conclude $||T\_{n}||$ is uniformly bounded? Another question if a linear functional $\varphi$ is continuous in unit ball in $M$ in weak operator topology does it imply that $\varphi$ is normal that means in the predual?



On predual of von Neumann algebra
Suppose $M$ acts on a Hilbert space $H$, for a fixed $x\in H$, the set $\{\langle T\_nx,y\rangle: n\in\mathbb N\}$ is bounded for all $y\in H$, so $\{T\_nx: n\in\mathbb N\}$ is bounded by the principle of uniform boundedness, and again using principle of uniform boundedness we conclude that $\{T\_n: n\in\mathbb N\}$ is also bounded. 


Yes to the second question, see for example Theorem 1.10 in "lectures on von Neumann algebras" by "S. Strătilă, L. Zsido". 


