Continuity of the Fenchel correspondence and continuity of polarities (Q1176894)

The polar cone of the limit inferior of a sequence of cones of a Banach space is shown to be Cesari's limit superior of the sequence of polar cones in the bounded weak\(^*\) topology. In a similar way the lower semicontinuity of a not necessarily countable family of closed convex cones is characterized in terms of Castaing's notion of pseudo-upper semicontinuity. More general results are given within the framework of convergence spaces and the pseudo-upper semicontinuity of a closed-valued multifunction is characterized as closedness with respect to a new convergence. Applications to the Young-Fenchel correspondence are pointed out.