Abstract
A collection X Λ = {x α : α ∈ Λ} of nonzero elements of a complete separable locally convex space H over a scalar field Ψ (Ψ = ℝ or ℂ), where Λ is a certain set of subscripts, is said to be an absolutely representing family (ARF) in H if ∀ x ∈ H one can find a family in the form {c α x α : c α ∈ Ψ, α ∈ Λ} which is absolutely summable to x in H. In this paper we study certain properties of ARF in Fréchet spaces and strong adjoints to reflexive Fréchet spaces. We pay the most attention to obtaining the criteria that allow one to conclude that a given collection X Λ is an ARF in H.
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Dedicated to the memory of Pyotr Lavrent’evich Ul’yanov
Original Russian Text © Yu.F. Korobeinik, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 9, pp. 25–35.
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Korobeinik, Y.F. The absolutely representing families in certain classes of locally convex spaces. Russ Math. 53, 20–28 (2009). https://doi.org/10.3103/S1066369X09090035
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DOI: https://doi.org/10.3103/S1066369X09090035