1. See his forthcoming article, ‘Quantum Mechanics and Hilbert Space’, Amer. Math. Monthly.
2. The present version of this lemma and its proof are due to R. S. Palais, who was kind enough to read the first draft of this paper.
3. This lemma is due to Jordan and von Neumann, On inner products in linear metric space, Annals of Math. 36 (1935), pp. 719–723.