I was reading the folowing text about proving certain map between Banach spaces is smooth. 
The group under consideration is the gauge group (howver I do not think it is essential here). We are trying to prove that the multiplication is smooth, while it is continuous by Sobolev multiplication theorem, I wonder how we then get the smoothness of the map?
p.s. I'm not quite familiar with Sobolev theory so I apologize if that is a stupid question.