Semidifferential calculus (Q1085397)

Let \(X\) and \(Y\) denote topological vector spaces (TVS's). In this paper we present a theory of differentiation in which the derivative (called by us semiderivative) \(f'[x_ 0]\) of \(f: X\to Y\) at \(x_ 0\) is a sequentially continuous positively homogeneous map from X to Y which is not necessarily additive. Functions \(f: X\to Y\) for which all possible one- sided directional derivatives exist are semidifferentiable in the weakest (Gâteaux) sense when these directional derivatives depend sequentially continuously on the ''directions''. We also discuss Hadamard and Fréchet semiderivatives, in the now customary manner using uniform limits on the sets \(\Sigma\) of a cover of X, due to Sebastião e Silva. The first section, {\S} 1, discusses the basic concepts. In (1.9) we generalize a theorem of Vainberg asserting the linearity of \(f'[x_ 0]\) when the map \(x\to f'[x]\) is continuous at \(x_ 0\). {\S} 2 is concerned with the fundamental theorem of calculus and a slight generalization of an important iterative fixed point result. In {\S} 3, we discuss computational rules, especially the composite and derivative-of-the- inverse theorems. Some examples are given in {\S} 4. Finally, in {\S} 5, we make several comments about generalizations.