Differential Equations in Abstract Spaces (Enhanced Edition‪)‬

In this preliminary chapter the reader will be familiarized with those
parts of the calculus of abstract functions that are essential in the study of
differential equations in Banach and Hilbert spaces. By an abstract function
we mean a function mapping an interval of the real line into a Banach space.
We begin by defining weak and strong continuity and differentiability of
abstract functions and prove a form of the mean value theorem for abstract
functions. Next we develop the Riemann integral for abstract functions and
those properties of this integral which are constantly used in the text. We
then outline abstract integrals of the Lebesgue type (Pettis and Bochner
integrals) and state some basic results. We also sketch the abstract Stieltjes
integral for functions mapping a Banach space into another Banach space.
Finally we treat in some detail the Gateaux and Frtchet differential of
functions mapping a Banach space into another Banach space.