The dual stuff is just a framework for thinking about how you're avoiding multiplies at display-time.
Operators are a layer of notation that makes it easier to work with nasty integral equations.
Norms tell su how big things are. The operator M takes radiance functions to radiance functions. It's often nice to split an operator up into two other operators. K deals with an interaction at a specific point. It takes everything coming in and weights it by the reflectivity at a point.