According to theorem 7 and proposition 6, an alternate proof consists in showing that p is ?(X * , X )-lower semi-continuous. The following proof is a slight generalization of the one of ,
, Also denote by ? : X ?X the canonical projection. By Hahn-Banach theorem, {0} = {x ? X | ?? ? X * ?(x) = 0}, so the topological dual ofX is isomorph to X * : we will identify both, writing, for (?, x) ? X * × X , ?(x) = ?(?(x)). Moreover, we have ?(X * , X ) = ?(X * ,X ), First, denote by (X ,? ) the Hausdorff topological vector space canonically associated to
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