The Representation Theorems of Conjugate Spaces of Some $l^{0}(\{X_{i}\})$ Type $F$-Normed Spaces
Received:August 11, 2016  Revised:November 11, 2016
Key Words: $l^{0}(\{X_{i}\})$ type $F$-normed space   locally convex space   locally bounded space   sequential weak star topology   representation theorem
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11471236).
 Author Name Affiliation Jianyong WANG Department of Mathematics, Changshu Institute of Technology, Jiangsu 215500, P. R. China
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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.