A new proof of a lemma by Phelps

This lemma was then used the following year as a crucial step in the proof of the well-known Bishop-Phelps theorem [1] that every Banach space is subreflexive; in other words, every functional on a Banach space E can be approximated by a norm-attaining functional on the same space. The original proof of this lemma uses the Hahn-Banach theorem and is therefore fairly abstract. In this note, we present an alternate proof for Lemma 1 stated above. This proof gives a geometric argument while extending the lemma to a complex Banach space. Lemma 1 is shown to be a special case when the bound of on ‖ f ± g‖ is replaced by 5 . This replacement does not affect the fundamental conclusion of Lemma 1. We now state the extended lemma.