## Abstract

We introduce a notion which is intermediate between that of taking the w*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example: 1. Assume X is separable and non-reflexive and its unit sphere is covered by a sequence of balls {C _{i}}_{i=1}^{∞} of radius a 1. Then for every sequence of positive numbers {ε_{i}}_{i=1} ^{∞} tending to 0 there is an f ∈ X*, such that ∥ f ∥ = 1 and f(x) ≤ 1 - ε_{i}, whenever x ∈ C _{i}, i = 1,2,.. .. 2. Assume X is separable and non-reflexive and let T: Y → X* be a bounded linear non-surjective operator. Then there is an f ∈ X* which does not attain its norm on B_{X} such that f ∉ T (Y).

Original language | English |
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Pages (from-to) | 157-172 |

Number of pages | 16 |

Journal | Israel Journal of Mathematics |

Volume | 136 |

DOIs | |

State | Published - 1 Jan 2003 |

## ASJC Scopus subject areas

- Mathematics (all)