It starts from the existence of the lower semi-continuous convex functional extremum in the reflexive space

and moves on to the non-reflexive space. Some conclusions were drawn

based on the weak star lower semi-continuous property

or the weak star lower semi-continuous from above property of convex function in the non-reflexive space. These results were applied to the lagrangian convex function. It draws the conclusion that the lagrangian convex function also has the self-dual property.