A Simple Norm Inequality

On sharp triangle inequalities in Banach spaces

“…The triangle inequality is one of the most fundamental inequalities in analysis and have been treated by many authors (e.g., [1][2][3][8][9][10], etc.). Recently Kato, Saito and Tamura [5] showed the following sharp triangle inequality and its reverse inequality with n elements in a Banach space.…”

The Dunkl–Williams constant, convexity, smoothness and normal structure

“…We can assume, passing to subsequences if necessary, that the limits lim n→∞ f n (x n ) and lim n→∞ f n + f exist, and also that {f n } converges in the weak* topology to some f * ∈ X * . Taking into account (3), that x n w − → 0, that f n w * − − → f * and also that f * (x n ) → 0, we may assume in addition, via a standard procedure, that …”

On the calculation of the Dunkl-Williams constant of normed linear spaces

“…[3,5,6,13,16,18,19]. In [8], it has been proved that the constant 4 can be replaced by 2 if X is an inner product space, and also that the value 4 is the best possible choice in the space (R 2 · 1 ). A bit later, Kirk and Smiley [14] showed that a normed linear space X is an inner product space if the inequality − ≤ 2 − + holds for any nonzero elements in X .…”