Strong Subdifferentiability of Convex Functionals and Proximinality

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Using strong subdifferentiability of convex functionals, we give a new sufficient condition for proximinality of closed subspaces of finite codimension in a Banach space. We apply this result to the Banach space K(l2) of compact operators on l2 and we show that a finite codimensional subspace Y of K(l2) is strongly proximinal if and only if every linear form which vanishes on Y attains its norm.