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THE DIRECT LIMITS OF THE BANACH–MAZUR COMPACTA

Published online by Cambridge University Press:  21 December 2000

TARAS BANAKH ,
KAZUHIRO KAWAMURA  and
KATSURO SAKAI
TARAS BANAKH
Affiliation:
Department of Mathematics, Lviv State University, Lviv, 79000, Ukraine; e-mail: [email protected]
KAZUHIRO KAWAMURA
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan; e-mail: [email protected], [email protected]
KATSURO SAKAI
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan; e-mail: [email protected], [email protected]
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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 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.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 32 , Issue 6 , November 2000 , pp. 709 - 717
Copyright
© The London Mathematical Society 2000

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