OAI.LipschitzCounterexample.main

The theorem states that the defined proposition MainClaim holds (it is admitted in the source, not proved here). MainClaim asserts that there exist two separable real Banach spaces X and Y, each given as a type with a norm, a real normed-space structure, completeness and separability, together with a bijection Ψ from X onto Y, such that Ψ is a bi-Lipschitz equivalence with explicit constants: for all s and t in X, (4/21)‖s−t‖ ≤ ‖Ψ(s)−Ψ(t)‖ ≤ (76/25)‖s−t‖. Moreover, there is no continuous linear equivalence (linear homeomorphism) between X and Y, so the two spaces are Lipschitz equivalent but not linearly isomorphic. In addition, there is a linear isometric embedding of C0L2 into X, where C0L2 is the space of continuous functions from the natural numbers to the real Hilbert space ℓ² that vanish at infinity. Finally, Y does not contain a linear copy of C0L2, meaning there is no bounded linear map T from C0L2 to Y and constant a>0 with a‖x‖ ≤ ‖T x‖ for every x in C0L2.