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<identifier>oai:HAL:hal-00699215v1</identifier>
<datestamp>2017-12-21</datestamp>
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<publisher>HAL CCSD</publisher>
<title lang=en>Hahn-Banach theorems for convex functions</title>
<creator>Lassonde, Marc</creator>
<contributor>Laboratoire de Mathématiques Informatique et Applications (LAMIA) ; Université des Antilles et de la Guyane (UAG)</contributor>
<description>International audience</description>
<source>Nonconvex Optim. Appl.</source>
<source>Minimax theory and applications</source>
<coverage>Erice, Italy</coverage>
<identifier>hal-00699215</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00699215</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00699215/document</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00699215/file/M_NonconvexOptimAppl98.pdf</identifier>
<source>https://hal.archives-ouvertes.fr/hal-00699215</source>
<source>Minimax theory and applications, Sep 1996, Erice, Italy. 26, pp.135-145, 1998</source>
<language>en</language>
<subject>[MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA]</subject>
<type>info:eu-repo/semantics/conferenceObject</type>
<type>Conference papers</type>
<description lang=en>We start from a basic version of the Hahn-Banach theorem, of which we provide a proof based on Tychonoff's theorem on the product of compact intervals. Then, in the first section, we establish conditions ensuring the existence of affine functions lying between a convex function and a concave one in the setting of vector spaces -- this directly leads to the theorems of Hahn-Banach, Mazur-Orlicz and Fenchel. In the second section, we caracterize those topological vector spaces for which certain convex functions are continuous -- this is connected to the uniform boundedness theorem of Banach-Steinhaus and to the closed graph and open mapping theorems of Banach. Combining both types of results readily yields topological versions of the theorems of the first section.</description>
<date>1996-09-30</date>
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