In any dual space X*, the set QP of quasi-polyhedral points is contained in the set SSD of points of strong subdifferentiability of the norm which is itself contained in the set NA of norm attaining functionals. We show that NA and SSD coincide if and only if every proximinal hyperplane of X is strongly proximinal, and that if QP and NA coincide then every finite codimensional proximinal subspace of X is strongly proximinal. Natural examples and applications are provided.

Let $X$ be a finite dimensional Banach space and let $Y\subset X$ be a hyperplane. Let $\text{L}{}_Y=\{L\in \text{L}(X,Y):L{\mid }_Y=0\}$. In this note, we present sufficient and necessary conditions on $L_0\in \text{L}{}_Y$ being a strongly unique best approximation for given $L\in \text{L}\left(X\right)$. Next we apply this characterization to the case of $X=l_{\infty }^n$ and to generalization of Theorem I.1.3 from [12] (see also [13]).

We characterize strongly proximinal subspaces of finite codimension in C(K) spaces. We give two applications of our results. First, we show that the metric projection on a strongly proximinal subspace of finite codimension in C(K) is Hausdorff metric continuous. Second, strong proximinality is a transitive relation for finite-codimensional subspaces of C(K).

En utilisant à la fois la théorie des polynômes orthogonaux et des arguments élémentaires de géométrie des nombres, nous donnons ici des nouveaux encadrements pour le diamètre transfini entier d’un intervalle $I$ d’extrémités rationnelles. Ces encadrements dépendent explicitement de la longueur de $I$ et des dénominateurs de ses extrémités.

We construct k-dimensional (k ≥ 3) subspaces $V^k$ of $l_1$, with a very simple structure and with projection constant satisfying $\lambda \left(V^k\right)\ge \lambda (V^k,l_1)>\lambda \left(l_2^{\left(k\right)}\right)$.