ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 46 Haller, Rainis, Tartu University, Estonia M(r, s)-inequality and reflexivity Let r ≤ 1 and s be positive numbers. We prove that a Banach space X satisfies the M(r, s)-inequality (i. e. ∥x^{∗∗∗}∥ ≥ r∥πx^{∗∗∗}∥ + s∥x^{∗∗∗} − πx^{∗∗∗}∥ ∀x^{∗∗∗} ∈ X^{∗∗∗}, where π is the canonical projection of X^{∗∗∗} onto X^{∗}) if and only if the unit sphere S_{X} of X has the following property: whenever ϵ > 0, x ∈ S_{X}, and (x_{n})_{n=1}^{∞} ⊂ S_{X}, then there are n_{0} ∈ ℕ, u ∈ conv{x_{1}, …, x_{n_{0}}}, and v ∈ conv{x_{n_{0}+1}, x_{n_{0}+2}, …} satisfying ∥rx + s(v − u)∥ ≤ 1 + ϵ. This immediately implies a well-known characterization of reflexivity due to R. C. James, generalizes criteria of M-ideals, and shows that the M(r, s)-inequality is separably determined. These results are joint work with E. Oja. Reference: R. Haller and E. Oja, Geometric characterizations of positions of Banach spaces in their biduals, Arch. Math. 69 (1997), 227–233.