On the characteristic of sets in the space conjugate to a~normed structure

V. A. Shestakov

Matematičeskie zametki, Tome 15 (1974) no. 6, pp. 915-924.

Résumé

Let $(E,\|\cdot\|_E)$ be a normed space, $E^*$ its conjugate, and M a linear subset in $E^*$. The number $$ r(M,E,\|\cdot\|_E)=\inf_{\substack{x\in E\\x\ne0}}\sup_{\substack{f\in M\\\|f\|\le1}}\frac{|f(x)|}{\|x\|_E} $$ 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. $(X,\|\cdot\|_X)$ be a $KN$-space. Then in order that $\|\cdot\|_X$ be semicontinuous on $X$ it is necessary and sufficient that for each intervally-complete norm $p$ on $X$ the $(X,\|\cdot\|_X)^*\cup(X,p)^*$, i.e., the set of all functionals linear on $X$, simultaneously continuous with respect to both the norm $\|\cdot\|_X$ and the norm $p$, have characteristic one in the space $(X,\|\cdot\|_X)$.