3) E. Picard, Train d'analyse, 2d ed.vol. 1 p. 170.The present approach was announced in Abstract 62-2-190 of Bull.Amer.Math.Soc, where the phrase "hyperbolic metric" was used, because of the relation to Hilbert's hyperbolic geometries.(4) In the sense of G. Birkhoff, Lattice theory, rev.ed., Chap.XV.Interpretations of the usual Banach spaces as vector lattices are explained there.
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Garrett Birkhoff (1957) studied this question.