# Some Generic Results on Non-attaining Functionals

Borwein, Jonathan M. and Kortezov, Ivaylo (2001) Some Generic Results on Non-attaining Functionals. Special Issue of Set-valued Analysis on Well-posedness and Stability of Optimization Problems, 9 . pp. 35-47.

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We prove that (a) in a reflexive space, for any linearly bounded but unbounded closed convex subset the non-support functionals are a dense $G_\delta$ subset of the polar set, and (b) any non-semicoercive proper convex lsc [weak$^*$-lsc] function in a [dual] Banach space has a generic [dense $G_\delta$] set of $L^\infty$-perturbations which do not attain their infimum. We also characterize the proper convex functions that have inf-nonattaining $L^\infty$-perturbations.This results also in a criterion for reflexivity.