derivative of the norm of a linear mapping; linear differential equations in a Banach space; normed space; solution of the differential equation
In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.
 A. N. Kolmogorov S. V. Fomin: Foundations of the theory of functions and functional analysis. (Russian). Czech translation, Praha, 1975.
 J. Dieudonné: Foundations of modern analysis
. Academic Press, New York-London, 1960; Russian translation, Mir, Moscow. MR 0120319