Convex Extension of Lower Semicontinuous Functions Defined on Normal Hausdorff Space

We prove that any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space is canonically equivalent to a problem of minimization of a proper weak-star lower semicontinuous convex function defined on a weak-star convex compact subset of some dual Banach space. We establish the existence of a bijective operator between the two classes of functions which preserves problems of minimization.