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Remarks on the $Γ$-regularization of Non-convex and Non-semi-continuous Functions on Topological Vector Spaces

We show that the minimization problem of any non-convex and non-lower semi-continuous function on a compact convex subset of a locally convex real topological vector space can be studied via an associated convex and lower semi-continuous function $Γ\left( h\right) $. This observation uses the notion of $Γ$-regularization as a key ingredient. As an application we obtain, on any locally convex real space, a generalization of the Lanford III-Robinson theorem which has only been proven for separable real Banach spaces. The latter is a characterization of subdifferentials of convex continuous functions.