The Representation Theorems of Conjugate Spaces of Some $l^{0}(\{X_{i}\})$ Type $F$-Normed Spaces

Abstract:

      In a paper published in {\sl Acta Mathematica Sinica} (2016, 59(4)) we obtained some representation theorems for the conjugate spaces of some $l^{0}$ type $F$-normed spaces. In this paper, for a sequence of normed spaces $\{X_{i}\}$, we study the representation problems of conjugate spaces of some $l^{0}(\{X_{i}\})$ type $F$-normed spaces, obtain the algebraic representation continued equalities $$(l^{0}(\{X_{i}\}))^{*}\stackrel{A}{=}(c_{00}^{0}(\{X_{i}\}))^{*}\stackrel{A}{=}c_{00}(\{X^{*}_{i}\}),$$ $$(l^{0}(X))^{*}\stackrel{\mathrm{A}}{=}(c^{0}(X))^{*}\stackrel{\mathrm{A}}{=}(c_{0}^{0}(X))^{*}\stackrel{\mathrm{A}}{=}(c_{00}^{0}(X))^{*}\stackrel{\mathrm{A}}{=}c_{00}(X^{*}),$$ and the topological representation $((c^{0}_{00}(\{X_{i}\}))^{*},sw^{*})=c^{0}_{00}(\{X^{*}_{i}\})$, where $sw^{*}$ is the sequential weak star topology. For the sequences of inner product spaces and number fields with the usual topology, the concrete forms of the basic representation theorems are obtained at last.