Differentiation in Topological Vector Spaces

Differentiation in mathematical analysis is commonly built by using {\epsilon}-{\delta}-language. This approach also works similarly for defining continuity, Gateaux (directional) derivative and Frechet derivative in normed vector spaces, in particular, in Banach spaces, where Frechet derivatives are defined as limits of ratios with respect to the norms in the considered normed vector spaces. For general topological vector spaces, if the space is not equipped with a norm, then Frechet derivatives cannot be similarly defined as in normed vector spaces. The cornerstone of this paper is the fact that the topology of every topological vector space can be induced by a family of F-seminorms, which is used to develop an extended {\epsilon}-{\delta}-language with respect to the F-seminorms. By using the extended {\epsilon}-{\delta}-language in topological vector spaces, we first define the continuity of single-valued mappings. Then we define Gateaux and Frechet derivatives as a certain type of limits of ratios with respect to the F-seminorms equipped on the considered spaces, which are naturally generalized Gateaux and Frechet derivatives in normed vector spaces. We will prove some analytic properties of the generalized versions of Gateaux and Frechet derivatives, which are similar to the analytic properties in normed vector spaces. Then we apply them to some general topological vector spaces that are not normed, such as the Schwartz space and other two spaces that are not even locally convex. For some single-valued mappings defined on these three spaces, we will precisely calculate their Gateaux and Frechet derivatives. Finally, we apply the generalized Gateaux and Frechet derivatives to solve some vector optimization problems and investigate the order monotonic of single-valued mappings in general topological vector spaces.