. Proof, From Lemma 3, we have that f × (?) = inf ??A Y (f ) ? × (?), for all ? ? C b (X)

?. Sci, Let us prove thatX))) is a bijective map. Indeed, using the part (1) of Lemma 1, we get that T is one to one To see that T is onto, let g ? ?(co w * (?(X))), there exists a ?-set A such that g = (? A ) |co w * (?(X)) Using Proposition 2, there exist a bounded from below lower semicontinuous function f : X ?? R such that A = A Y (f ) := {? ? C b (X)/? ? f }. Thus, by using Proposition 3, we get that g = T (f ) i.e. T is onto, Now, we prove that for all f, g ? SCI(X) and all ?, ? ? R + , we have T (?f + ?g) = ?T (f ) + ?T (g)

L. Indeed and ?. +. , If ? = 0, it is easy to see that (?f ) × (?) = ?f × ( ? ? ) for all f ? SCI(X) Thus, ((?f ) × ) * = ?(f × ) * which implies that T (?f ) = ?T (f ) for all f ? SCI(X) On the other hand, if f, g ? SCI(X), then by applying Theorem 3, we get that (f + g) × = f × ? g ×, Hence by the properties of the Fenchel conjugacy, ((f + g) × ) * = (f × ? g × ) * = (f × ) * + (g × ) * . In other words, we have that T

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