Bounds for linear functionals on convex sets
Walter J. Roth · Mathematische Nachrichten · 2011
Abstract We consider continuous monotone linear functionals on a locally convex ordered topological vector space that are sandwiched between a given \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(\mathbb R\cup \lbrace +\infty \rbrace )$\end{document} ‐valued sublinear and an \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(\mathbb R\cup olinebreak \lbrace -\infty \rbrace )$\end{document} ‐valued superlinear functional. We review conditions for the existence of such functionals and in our main results investigate their range of suprema and infima on a given convex subset. These yield effective versions of the Hahn‐Banach theorem which give easy access to various applications including separation properties for convex sets, a non‐Baire approach to the Uniform Boundedness theorem, the notion of sub‐and superharmonicity with respect to a subcone, and results for the extension of monotone affine functions.