On some inequalities in normed linear spaces

Abstract

Upper and lower bounds for the norm of a linear combination of vectors are given. Applications in obtaining various inequalities for the quantities $ \Vert x / \Vert x \Vert -y / \Vert y \Vert \Vert $ and $ \Vert x/ \Vert y \Vert -y/ \Vert x \Vert \Vert $, where $ x $ and $ y $ are nonzero vectors, that are related to the Massera-Schaffer and the Dunkl-Williams inequalities are also provided. Some bounds for the unweighted Cebysev functional are given as well.