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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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