Asymptotic Almost Periodic Functions with Range in a Topological Vector Space

The notion of asymptotic almost periodicity was ？first introduced by Fréchet in 1941 in the case of ？finite dimensional range spaces. Later, its extension to the case of Banach range spaces and locally convex range spaces has been considered by several authors. In this paper, we have generalized the concept of asymptotic almost periodicity to the case where the range space is a general topological vector space, not necessarily locally convex. Our results thus widen the scope of applications of asymptotic almost periodicity. 1. Introduction The theory of almost periodic functions was mainly created and published during 1924–1926 by the Danish mathematician Harold Bohr. In 1933, Bochner [1] published an important article devoted to extension of the theory of almost periodic functions on the real line with values in a Banach space . His results were further developed by several mathematicians, see, for example [2–7]. The concept of asymptotic almost periodicity was first considered by Fréchet [8, 9] in 1941 for functions with the range restricted to a finite dimensional space. The semigroup case of turns out to be significantly different from the group case of . If is a Banach space or a locally convex case and replaced by , with , this notion has been extensively studied in recent years (see [10–14]). In this paper, we generalize the concept of asymptotic almost periodicity to the case of , a general topological vector space. 2. Preliminaries In this section, we give prerequisites on topological vector spaces and almost periodic functions for our main results of Section 3. Throughout this paper, denotes a nontrivial Hausdorff topological vector space (in short, a TVS) with a base of closed balanced shrinkable neighborhoods of . (A neighborhood of in is called shrinkable [15] if for .) By [15, Theorem 4 and 5], every Hausdorff TVS has a base of shrinkable neighborhoods of , and also the Minkowski functional of any such neighborhood is continuous and satisfies We mention that, for any neighborhood of in , need not be absolutely homogeneous or subadditive; however, the following useful properties hold [15, 16].(a) is positively homogeneous; it is absolutely homogeneous if is balanced.(b)If is a balanced neighborhood of in with , then A complete metrizable TVS is called an -space. Notations. Let be a completely regular Hausdorff space, and let be the set of all continuous functions -valued functions on . Furthermore, let Clearly, , , and all these sets are vector spaces over with the pointwise operations of addition and scalar multiplication. The uniform