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Linear topologies on sesquilinear spaces of uncountable dimension. (English) Zbl 0766.03027

Summary: F. Appenzeller [J. Symb. Logic 54, No. 3, 689-699 (1989; Zbl 0687.03030)] introduced an invariant \(\Gamma\) for orthosymmetric sesquilinear spaces of regular uncountable dimension \(\kappa\) which takes its values in some Boolean algebra \({\mathcal D}(\kappa)\). Constructively he shows that \(\Gamma\) maps onto \({\mathcal D}(\omega_ 1)\). We show that this is not true for \(\kappa>\omega_ 1\). Orthosymmetric sesquilinear spaces naturally bear linear topologies defined by the form. There are various relations between the arithmetic, geometric and topological properties of such spaces. E.g., W. Bäni [Commun. Algebra 5, 1561-1587 (1977; Zbl 0365.15002)] characterizes \(\gamma\)-diagonal spaces using the notions of \(\gamma\)-compactness and continuous bases. We present an alternative characterization: existence of a convergent algebraic basis. Bäni [loc. cit.] asked whether there exist \(\gamma\)-compact spaces without continuous bases for arbitrary regular \(\gamma\). We give a positive answer by showing that the spaces defined by Appenzeller are examples of this.

MSC:

03E35 Consistency and independence results
15A63 Quadratic and bilinear forms, inner products
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
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