Biconjugate Functions

As follows from the Fenchel–Moreau Theorem, when the dual of a normed space X is endowed with the weak∗ topology, the biconjugate of a proper, convex and lower semicontinuous function defined on X coincides with the function itself. This is not necessarily the case when X ∗ is endowed with the strong topology. Working in the latter setting, we give in this chapter formulae for the biconjugates for some classes of functions, which appear in the convex optimization, that hold provided the validity of some suitable regularity conditions. Consider X a normed space with the norm $$\|\cdot\|$$ and X ∗ its topological dual space, the norm of which being denoted by $$\|\cdot\|_{\ast}$$ . On this space we work with three topologies, namely the strong one induced by $$\|\cdot\|_{\ast}$$ which attaches X ∗∗ as dual to X ∗, the weak∗ one induced by X on X ∗, ω(X ∗, X), which makes X to be the dual of X ∗ and the weak one induced by X ∗∗ on X ∗, ω(X ∗, X ∗∗), that is the weakest topology on X ∗ which attaches X ∗∗ as dual to X ∗∗. We specify each time when a weak topology is used, otherwise the strong one is considered. Like in the previous section, for sets and functions the closures and the lower semicontinuous hulls, respectively, in the weak∗ topologies are denoted by clω*, while the ones in the weak topologies are denoted by clω. A normed space X can be identified with a subspace of X ∗∗, and we denote by $$\widehat{x}$$ the canonical image in X ∗∗ of the element $$x \in X,\ {\rm i.e.}\ \langle\widehat{x}, x^{\ast}\rangle = \langle x^{\ast}, x\rangle\ {\rm for\ all}\ x \in X\ {\rm and}\ x^{\ast} \in X^{\ast}$$ , where by $$\langle\cdot,\cdot\rangle$$ we denote the duality product in both $$X^{\ast} \times X\ {\rm and}\ X^{\ast\ast} \times X^{\ast}.\ {\rm For}\ U \subseteq X\ {\rm denote\ also}\ \widehat{U} = \{\widehat{x} : x \in U\}$$ . In this setting, when $$f : X \rightarrow \overline{\mathbb{R}}$$ is a given function, its biconjugate $$f^{\ast\ast} : X^{\ast\ast} \rightarrow \overline{\mathbb{R}}$$ is defined by $$f^{\ast\ast} (x^{\ast\ast}) = {\rm sup}\{\langle x^{\ast\ast}, x^{\ast}\rangle - f^{\ast}(x^{\ast}) : x^{\ast} \in X^{\ast}\}$$ . For the beginning we prove the following preliminary result.