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Geometric characterizations of positions of Banach spaces in their biduals. (English) Zbl 0914.46019

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.

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

Citations:

Zbl 0823.46023
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