A note on Taylor's formula

Abstract． We show that Taylor，s formula given in［2］holds． under a more genera1 setting， in particular， that in［2， Theorem 1］， the assumption of the R-v-derivative of order n being symmetric is needless． 1991Mat九ematics 5勉句ec2 classification．46A99，58C20． Keyωords． Ranked space， Non-syrmnetric preneighborhood， Taylor，s for-mula． 0．Introduction Nagakura［8］investigated differential calculus in lineall ra皿ked spaces with non-symlhetric preneighborhoods(see also［9］)， and in［2］，［3］ and［4］ ．we 高≠р?@a further study of the calculus in such spaces． The method taken there， i．e． that of ranked spaces(cf．［5］，［6D， is consideral｝ly different from those of other theories of differentia1 calculus ip li n．ear spaceS． It seems to us that the results of［2］，［3］，［4］，［8］and［9］have a wide range of applica-