关键词: l⁰  ({X_i})type F-normed space, locally convex space, locally bounded space, conjugate space, representation theorem

Abstract: Extending the results of an article published in(Acta Mathematica Sinica（2016,59（4）） by the author, for a sequence of normed spaces {X_i}, the representation problem of conjugate spaces of some l⁰（{X_i}） type F-normed spaces are studied in this paper. The algebraic representation continued equalities l⁰（{X_i}） * A=c₀₀⁰（{X_i}） * A= c₀₀（{X_i*}）,（l⁰（X））* A=（c⁰（X） ）* A=（c₀⁰（X））* A=(c₀₀⁰（X）)* A= c₀₀（X*）are obtained in the first part. Under weak-star topology, the topological representation c₀₀⁰（{X_i}） *, w* = c₀₀⁰（{X_i*}） is obtained in the second part. For the sequence of inner product spaces and number fields with the usual topology, the concrete forms of the basic representation theorems are obtained at last. 

Key words: l⁰  ({X_i})type F-normed space, locally convex space, locally bounded space; conjugate space, representation theorem