(I)-envelopes of unit balls and James' characterization of reflexivity

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We study the (I)-envelopes of the unit balls of Banach spaces. We show, in particular, that any nonreflexive space can be renormed in such a way that the (I)-envelope of the unit ball is not the whole bidual unit ball. Further, we give a simpler proof of James' characterization of reflexivity in the nonseparable case. We also study the spaces in which the (I)-envelope of the unit ball adds nothing. 1. Introduction. The present paper is a continuation of (13) where the (I)-envelopes of sets in Banach spaces were introduced and studied with the aim of using the result of (8) to obtain an easier proof of James' charac- terization of weak compactness in nonseparable spaces. It was shown there that an easy proof can be given in a large class of spaces. However, an example was exhibited showing that the general situation is not so easy ((13, Example 4.1)). In this paper we study the (I)-envelopes of the unit balls and in particu- lar two extreme classes of spaces: those where the (I)-envelope adds nothing (Section 4) and those where the (I)-envelope is as large as it can be (Sec- tions 2 and 3). In Section 3 we give a simpler proof of James' characterization of reflexivity using recent results of (15). Let us start by recalling the definition of the (I)-envelope and its basic properties. Let X be a Banach space and B ⊂ X ∗ . The (I)-envelope of B is defined