Let 1 [les ] p [les ] ∞. For each n-dimensional Banach space E = (E, ∥ · ∥), we define a norm ∥ · ∥p on E × ℝ as follows:

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It is shown that the correspondence (E, ∥ · ∥) [map ] (E × ℝ, ∥ · ∥p) defines a topological embedding of one Banach–Mazur compactum, BM(n), into another, BM(n + 1), and hence we obtain a tower of Banach–Mazur compacta: BM(1) ⊂ BM(2) ⊂ BM(3) ⊂ ···. Let BMp be the direct limit of this tower. We prove that BMp is homeomorphic to Q∞ = dir lim Qn, where Q = [0, 1]ω is the Hilbert cube.