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Normed barrelled spaces. (English) Zbl 0974.46003

Przeworska-Rolewicz, Danuta (ed.), Algebraic analysis and related topics. Proceedings of the conference, Warsaw, Poland, September 21-25, 1999. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 53, 205-210 (2000).
Let \(X\) be a normed space. A set \(S\subset \text{Sphere}(X)\) is called a bounding set if for each unbounded sequence \((f_n)\) in the dual space \(X'\) there exists a sequence \((x_n)\) in \(S\) such that \(\sup_n|f_n(x_n)|= \infty\). The main result of the paper (Theorem) represents the following sufficient condition under which \(X\) is barrelled:
For any sequence \((x_n)\) in \(S\) there exist a sequence \((d_k)\in \text{Ball}(\ell^1)\) with \(d_k\neq 0\) and integers \(N_k\geq 0\), \(C> 0\), such that for every subsequence \((x_{n_k})\) of \((x_n)\) there is a further subsequence \((x_{n_{k_\ell}})\) and \(x\in X\) with \(x= \sum^\infty_{j=1} t_jx_j\) and \[ \|t_{n_{k_\ell}} x_{n_{k_\ell}}\|- \sum_{\substack{ j= n_{k_{\ell-1}}+ N_{k_{\ell-1}}+ 1\\ j\neq n_{k_\ell}}} \|t_j x_j\|\geq C|d_{k_\ell}|. \] From this theorem, the author obtains that the space of Pettis integrable functions is barrelled [cf. L. Drewnowski, M. Florencio and P. J. Paúl, Proc. Am. Math. Soc. 114, No. 3, 687-694 (1992; Zbl 0747.46026)]. Furthermore, it is shown that the main result contains a series of known theorems concerning the barrelledness of subspaces of \(\ell^1\).
For the entire collection see [Zbl 0944.00033].

MSC:

46A08 Barrelled spaces, bornological spaces
46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0747.46026
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