The Direct Limits Of The Banach-mazur Compacta

Abstract

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 BM p be the direct limit of this tower. We prove that BM p is homeomorphic to Q ∞ = dir lim Q n , where Q = [0, 1] ω is the Hilbert cube.