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