Abstract: 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.

Key words: weak star lower semi-continuous; weak star lower semicontinuous from above; self-dual lagrangian convex function