Resumen de Semidifferential calculus
David F. Findley
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 (G\^{a}teaux) 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 Sebasti\~{a}o e Silva (cf. [2], [3], [18], [20], [23]).\newline 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. For deeper versions of some of the results of section 3 in more restricted contexts, see [25] and [26].
