Haller, Rainis; Oja, Eve Geometric characterizations of positions of Banach spaces in their biduals. (English) Zbl 0914.46019 Arch. Math. 69, No. 3, 227-233 (1997). Summary: Let \(r\leq 1\) and \(s\) be positive numbers. We prove that a Banach space \(X\) satisfies the \(M(r,s)\)-inequality (i.e. \[ \| x^{***}\|\geq r\|\pi x^{***}\|+ s\| x^{***}- \pi x^{***}\|\qquad \forall x^{***}\in X^{***}, \] where \(\pi\) is the canonical projection of \(X^{***}\) onto \(X^*\)) if and only if the unit sphere \(S_X\) of \(X\) has the following property: whenever \(\varepsilon> 0\), \(x\in S_X\), and \((x_n)^\infty_{n=1}\subset S_X\), then there are \(n_0\in\mathbb{N}\), \(u\in \text{conv}\{x_1,\dots, x_{n_0}\}\), and \(v\in\text{conv}\{x_{n_0+ 1}, x_{n_0+2},\dots\}\) satisfying \[ \| rx+ s(v- u)\|\leq 1+\varepsilon. \] This immediately implies a well-known characterization of reflexivity due to R. C. James, giving its alternative proof based on Alaoglu’s and Goldstine’s theorems. This also generalizes criteria of \(M\)-ideals from [Å. Lima, E. Oja, T. S. S. R. K. Rao and D. Werner, Mich. Math. J. 41, 473-490 (1994; Zbl 0823.46023)], and shows that the \(M(r,s)\)-inequality is separably determined. Cited in 1 ReviewCited in 1 Document MSC: 46B20 Geometry and structure of normed linear spaces 46B10 Duality and reflexivity in normed linear and Banach spaces 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces Keywords:\(M(r,s)\)-inequality; reflexivity; Alaoglu’s and Goldstine’s theorems; \(M\)-ideals Citations:Zbl 0823.46023 PDFBibTeX XMLCite \textit{R. Haller} and \textit{E. Oja}, Arch. Math. 69, No. 3, 227--233 (1997; Zbl 0914.46019) Full Text: DOI