The parameters W(ε) and normal structure under norm and weak topologies in Banach spaces
✍ Scribed by Ji Gao
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 396 KB
- Volume
- 47
- Category
- Article
- ISSN
- 0362-546X
No coin nor oath required. For personal study only.
✦ Synopsis
Let (X) be a Banach space, and (S(X)) be the unit sphere of (X). Two geometric concepts are introduced: the modulus (W(\varepsilon)=\inf \left{\sup \left{, f_{x} \in \nabla_{x}\right}, x, y \in S(X)\right.) with (|x-y| \geq \varepsilon} ;) and the modulus (W_{1}(\varepsilon)=\sup \left{\inf \left{, f_{x} \in \nabla_{x}\right}, x, y \in\right.) (\mathrm{S}(\mathrm{X})) with (|\mathrm{x}-\mathrm{y}| \leq \varepsilon}) where (\nabla_{\mathrm{x}}) denotes the set of norm 1 supporting functionals of (\mathrm{S}(\mathrm{X})) at (\mathrm{x}, 0 \leq \varepsilon \leq 2). The properties of (\mathrm{W}(\varepsilon)) and (\mathrm{W}_{1}(\varepsilon)) and the relationship between these parameters and other geometric concepts under norm and weak topologies are discussed. The main results are that (\mathrm{W}(1)>0), or (\mathrm{W}(2)>1 / 2) and (\mathrm{W}_{1}(\varepsilon)<\varepsilon / 2) for any (\varepsilon, 0 \leq \varepsilon \leq 2) imply uniform normal structure.