INTRODUCTION

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

differentiable maps.

Except for the last chapter, we shall consider only veal locally convex

spaces.