We wish to construct a theory of differentiation in locally convex spaces
which has at least the following three properties: first, it must be in a
simple form so that using this theory does not require a comprehensive and
detailed knowledge about the theory of locally convex spaces; secondly, it must
be applicable to a fairly wide range of differential problems in analysis and
topology and, thirdly, it must have theorems which correspond to all important
theorems in Banach space calculus. At present, we do not have such a theory.
In fact, it has become clear that we can not have such a theory within the
traditional frame of the theory of locally convex spaces. If the aim of the
theory of locally convex spaces is to study such properties that are invariant
under topological linear transformations, then any theory of differentiation
with those three properties can not belong to the locally convex space theory.
The theory developed here depends on the choice of sets of semi-norms which
defines the locally convex topology of the space. Different sets of semi-norms,
still defining the same locally convex topology, may produce different sets of
Except for the last chapter, we shall consider only veal locally convex