Wang, Jianhua; Zhang, Zhihou Characterizations of the property \((C\)-\(K)\). (Chinese. English summary) Zbl 0917.46013 Acta Math. Sci. (Chin. Ed.) 17, No. 3, 280-284 (1997). Summary: In the theory of the best approximations the properties (\(C\)-\(K\)), \(K= \text{I,II,III}\) play a very significant role [cf. the first author, Math. Appl. 8, No. 1, 80-84 (1995)]. The purpose of this paper is to give their interesting characterizations. Further, we obtain that if a Banach space \(X\) has the property (\(C\)-III) (resp. (\(C\)-II)), then for any norming set \(A\) of \(X\), the \(\sigma(X,A)\) topology and the weak topology (resp. norm topology) coincide on the unit sphere of \(X\) and \(X\) is nearly very rotund (resp. nearly strongly rotund). Cited in 8 Documents MSC: 46B20 Geometry and structure of normed linear spaces 41A50 Best approximation, Chebyshev systems Keywords:best approximations; nearly very rotund; nearly strongly rotund PDFBibTeX XMLCite \textit{J. Wang} and \textit{Z. Zhang}, Acta Math. Sci. (Chin. Ed.) 17, No. 3, 280--284 (1997; Zbl 0917.46013)