F. Galaz-Fontes (Proc. AMS., 1998) has established a criterion
for a subset of the space of compact linear operators from a reflexive and
separable space X into a Banach space Y to be compact. F. Mayoral (Proc.
AMS., 2000) has extended this criterion to the case of Banach spaces not
containing a copy of l^1 . The purpose of this note is to give a new proof of the
result of F. Mayoral. In our proof, we use l^∞ -spaces, a well known result of
H. P. Rosenthal and L.E. Dor which characterizes the spaces without a copy
of l^1 and a recent result obtained by G. Nagy in 2007 concerining compact
sets in normed spaces. We point out that another proof of Mayoral’s result
was given by E. Serrano, C. Pineiro and J.M. Delgado (Proc. AMS., 2006) by
using a different method.