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The locally convex \(\mathcal A\)-spaces and their dual spaces. (English) Zbl 1023.46005
Let \((E,\tau)\) be a locally convex Hausdorff space. A sequence \((x_n)\) in \(E\) is said to be \(\tau\)-\(\mathcal K\)-convergent if each subsequence of \((x_n)\) has a further subsequence \((x_{n_k})\) such that the series \(\sum_{k=1}^{\infty}x_{n_k}\) is \(\tau\)-convergent to an element \(x\in E\). A subset \(B\) of \(E\) is said to be \(\tau\)-\(\mathcal K\)-bounded if for each sequence \((x_n)\) in \(B\) and each zero-sequence \((t_n)\) of scalars, the sequence \((t_nx_n)\) is \(\tau\)-\(\mathcal K\)-convergent.
Following Li Ronglu and C. Swartz [Stud. Sci. Math. Hung. 27, 379-384 (1992; Zbl 0681.46001)], the authors call the space \((E,\tau)\) an \(\mathcal A\)-space if each \(\tau\)-bounded subset of \(E\) is \(\tau\)-\(\mathcal K\)-bounded. They prove that if \((E,\tau)\) is an \(\mathcal A\)-space and \(E^s\) its sequentially dual space, then all the spaces \((E,\sigma(E,E^s))\), \((E,\beta(E,E^s))\), \((E^s,\sigma(E^s,E))\), \((E^s,\beta(E^s,E))\) are also \(\mathcal A\)-spaces. Particularly, if \(E\) is also a Mazur space (i.e., if \(E'=E^s\)), then \((E',\sigma(E',E))\) and \((E',\beta(E',E))\) are \(\mathcal A\)-spaces.
MSC:
46A03 General theory of locally convex spaces
46A32 Spaces of linear operators; topological tensor products; approximation properties
46A20 Duality theory for topological vector spaces
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