Abstract
Let (E, ∥ · ∥E) be a normed space, E* its conjugate, and M a linear subset in E*. The number is called the characteristic of the set M. In this paper we establish a relationship in normed structures between the semicontinuous properties of the norm and the characteristics of certain subsets in the conjugate space. For example, the following is a valid proposition. Let (X, ∥ · ||X) be a KN-space. Then in order that ∥ · ∥X be semicontinuous on X it is necessary and sufficient that for each intervally-complete norm p on X the set (X, ∥ · ∥X)* ∩ (X, p)*, i.e., the set of all functionals linear on X, simultaneously continuous with respect to both the norm ∥ · ∥X and the norm p, have characteristic one in the space (X, ∥ · ∥X).
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