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METRICS ON THE SEQUENCE SPACE Λp (f)

For 1 ≤ p, r < +∞ ,f( � 0) ∈ Lp(R ,d x) ,a ndg(� 0) ∈ Lr (R ,d x), the sequence spacep(f ) with metric d f p (a, b) was introduced in a previous paper and we discussed the inclusion relations betweenp andp(f ), and the linearity ofp(f ). The purpose of this paper is to discuss the topological structures of (� p(f ), d f p ). First we show that the space (� p(f ), d f ) is a complete separable metric group. Next we show that ifp(f ) is a linear space, then (� p(f ), d f p ) is a topologicallinear space. On the other hand, we give a necessary and sufficient condition for the inclusionp(f ) ⊂ � r (g). Furthermore, we show that the inclusions among the sequence spacesp(f ), � r (g) andr are continuous. The fact that � p(f ) ⊂ � r (g) as sets implies the continuity of the inclusion (� p(f ), d f p ) → (� r (g), d g r )