Let $f_n$ and $g_n$ two generalized functions such that :

• the product $f_n g_n$ is well defined for all $n\in \mathbb{N}$, which means $WF(f_n)+WF(g_n)$ does not contain an element of the form $(x,0)$.
  ( where $WF(u)$ is the wavefront of generalized function $f$)
• $f_n$ converges to $f$ and $g_n$ converges to $g$ (Weak* topologie).
• the product $f g$ is well defined.
  My question is : Can we deduce that $f_n g_n$ converge to $f g$ (weak* topologie) ?

Let $f_n$ and $g_n$ two generalized functions such that :

• the product $f_n g_n$ is well defined for all $n\in \mathbb{N}$, which means $WF(f_n)+WF(g_n)$ does not contain an element of the form $(x,0)$.
  ( where $WF(u)$ is the wavefront of generalized function $u$)
• $f_n$ converges to $f$ and $g_n$ converges to $g$ (Weak* topologie).
• the product $f g$ is well defined.
  My question is : Can we deduce that $f_n g_n$ converge to $f g$ (weak* topologie) ?