Nuclear and reflexive properties of some locally convex topologies on C(X)

Abstract

For a nonempty family \(\lambda \) of nonempty bounded subsets of a topological space X, \(C_{\lambda , u}(X)\) denotes the space C(X), of all real-valued continuous functions on a Tychonoff space X, equipped with the uniform topology generated by the family \(\lambda \); and it is a locally convex space. The aim of this paper is to study the situations when \(C_{\lambda , u}(X)\) can be a special kind of locally convex space such as a nuclear space, a Schwartz space or a semi-reflexive space. We also investigate when it is a reflexive or Montel space.