Metrics on the sequence space λ_p(f)

摘要

For 1 ≤ p, r <+∞, f(≠ 0) ∈ L_p(?, dx), and g(≠ 0) ∈ L_r (?, dx), the sequence space Λ_p(f) with metric d~f _p (a, b) was introduced in a previous paper and we discussed the inclusion relations between lp and Λ_p(f), and the linearity of Λp(f). The purpose of this paper is to discuss the topological structures of (Λ_p(f), d~f _p). First we show that the space (Λp(f), d~f _p) is a complete separable metric group. Next we show that if Λp(f) is a linear space, then (Λp(f), d~f _p) is a topological linear space. On the other hand, we give a necessary and sufficient condition for the inclusion Λ_p(f) ? Λr (g). Furthermore, we show that the inclusions among the sequence spaces Λp(f), Λr (g) and lr are continuous. The fact that Λp(f) ? Λr (g) as sets implies the continuity of the inclusion (Λp(f), d~f _p)→(Λr (g), dg _r) is emphasized.