OAI.C0Absorption.main_result

The theorem states that there exists a real normed space Z, in a fixed universe, with the following properties, where C₀ denotes the real Banach space of sequences ℝ-valued on ℕ that tend to zero at infinity, and a map f between metric spaces is bi-Lipschitz if there are constants 0<c≤C with c·d(x,y) ≤ d(f x,f y) ≤ C·d(x,y) for all x,y. First, Z is complete and separable. Second, Z contains no linear copy of C₀: every continuous linear map T from C₀ to Z fails to satisfy ‖Tx‖ ≥ c‖x‖ for all x for any c>0. Third, there is a surjective bi-Lipschitz map F from the product metric space Z × C₀ onto Z. Fourth, there is a bi-Lipschitz map from C₀ into Z. Fifth, Z is metrically universal for separable metric spaces: every separable metric space M, in the base universe, admits a bi-Lipschitz map into Z. Sixth, Z × C₀ is not linearly homeomorphic to Z, that is, there is no continuous linear equivalence, with continuous inverse, between Z × C₀ and Z.