Separation, convexity and polarity in the space of normlinear functions. (3rd April 2022) - Zaffaroni, A.
Abstract : The paper studies separation properties for subsets of the space N of normlinear functions on the Banach space X, i.e. the sum of a linear function and a multiple of the norm. We use as separating functionals the ones that are generated by the duality pairing [ x, p ] = p ( x ) = 〈 x, a 〉 + δ ‖ x ‖ . We show that normlinear functions can be used to separate points from radiant or coradiant sets in X . Then we pass to separation defined by means of the pairing [ x, ⋅ ], seen as a function defined on N . Since in this case the evaluation functionals are linear, separation describes a special subclass of convex sets in N . We characterize X -convex sets by exploiting the isomorphism between N and the space Y = X ∗ × R . We also study polarity relations defined on N . Polar sets are X -convex. And we describe what further properties are needed in order to make a subset C ⊂ Y the polar, or the reverse polar, of some set in X . For polar sets C ⊂ Y, it is possible to deduce the unique closed, radiant prepolar set U ⊂ X . To conclude, we emphasize some connections between upward (downward) X -convex sets in Y and sets of upper (resp. lower) bounds in Y .