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Linear topological properties of the Lumer-Smirnov class of the polydisc. (English) Zbl 0814.46018

Summary: Linear topological properties of the Lumer-Smirnov class \(LN_ *(\mathbb{U}^ n)\) of the unit polydisc \(\mathbb{U}^ n\) are studied. The topological dual and the Fréchet envelope are described. It is proved that \(LN_ *(\mathbb{U}^ n)\) has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for \(LN_ *(\mathbb{U}^ n)\).

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
46E25 Rings and algebras of continuous, differentiable or analytic functions
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