Upper and lower bounds for the norm of a linear combination of vectors
are given. Applications in obtaining various inequalities for the quantities ||x/||x||−
y/||y|||| and ||x/||y||−y/||x||||, 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.