Game-theoretic characterization of the Gurarii space

Wiesław Kubiś

doi:10.1007/s00013-017-1088-2

Abstract

We present a simple and natural infinite game building an increasing chain of finite-dimensional Banach spaces. We show that one of the players has a strategy with the property that, no matter how the other player plays, the completion of the union of the chain is linearly isometric to the Gurariĭ space.

Highlights

IntroductionWe shall denote this game by BM(B)
There exists a unique, up to linear isometries, separable Banach space G such that Odd has a strategy Σ in BM(B) leading to G, namely, the completion of every chain resulting from a play of BM(B) is linearly isometric to G, assuming Odd uses strategy Σ, and no matter how Eve plays
Research supported by GAC R grant 17-27844S and RVO 67985840 (Czechia). 1 game so that Odd uses his strategy leading to G1, while after the first move Eve uses Odd’s strategy leading to G2

Summary

Introduction

We shall denote this game by BM(B) This is a special case of an abstract Banach-Mazur game studied recently in [6]. There exists a unique, up to linear isometries, separable Banach space G such that Odd has a strategy Σ in BM(B) leading to G, namely, the completion of every chain resulting from a play of BM(B) is linearly isometric to G, assuming Odd uses strategy Σ, and no matter how Eve plays. Banach-Mazur game, Gurariı space, Eve. Research supported by GAC R grant 17-27844S and RVO 67985840 (Czechia). 1 game so that Odd uses his strategy leading to G1, while after the first move Eve uses Odd’s strategy leading to G2 Both players win, G1 is linearly isometric to G2. The Gurariı space is the unique object for which Odd has a winning strategy

Preliminaries

Proof of Theorem 1

Playing with a subclass of finite-dimensional spaces

Final remarks