# A Variation on the Banach-Dieudonne Theorem with Application to Maximal Monotone Operators

Eberhard, Andrew C. and Borwein, Jonathan M. (2007) A Variation on the Banach-Dieudonne Theorem with Application to Maximal Monotone Operators. (Unpublished)

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## Abstract

A convex set $C\subseteq X^{\ast}\times X$ is said to admit the Generalised Banach-Dieudonn\'{e} property (GBDP) if the weak$^{\ast}$-strong closure $\overline{C}^{w^{\ast}\times s}$ is characterised as the smallest set containing $C$ that is closed to all limits of its \emph{bounded} and weak$^{\ast}\times$s convergent nets. We show that all such convex sets $C\subseteq X^{\ast}\times X$ admit the GBDP when $X$ is either a separable dual Banach space or a reflexive Banach space. The situation in a general Banach space is more complicated, but we show that for a maximal monotone operator $T\subseteq X\times X^{\ast}$ with $0\in\func{int}\func{co} \func{Range}T$, when we use the embedding $\hat{T}\subseteq X^{\ast}\times X^{\ast\ast}$ then it admits representative functions whose epigraphs have the GBDP. Using the Lipschitz regularisation \cite{Borwein:1} this allows us to answer some outstanding problems regarding the embedding of Fitzpatrick and Penot representative functions for nonreflexive spaces. In particular, we show that the natural conjugate of the Fitzpatrick function, $\widehat{\mathcal{F}_{T}}^{\ast}:X^{\ast}\times X^{\ast\ast}\rightarrow \overline{\mathbf{R}}$ is itself a representative function when $T\subseteq X\times X^{\ast}$ is maximal monotone.

Item Type: Article pubdom FALSE Banach-Dieudonné theorem, monotone operators, representative functions 46-xx Functional analysis > 46Nxx Miscellaneous applications of functional analysis49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories46-xx Functional analysis > 46Axx Topological linear spaces and related structures47-xx Operator theory > 47Hxx Nonlinear operators and their properties UNSPECIFIED lingyun ye 29 Oct 2007 26 Feb 2015 15:14 https://docserver.carma.newcastle.edu.au/id/eprint/379