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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Absolute boundedness and absolute convergence in sequence spaces


Authors: Martin Buntinas and Naza Tanović-Miller
Journal: Proc. Amer. Math. Soc. 111 (1991), 967-979
MSC: Primary 40H05; Secondary 42A16, 46A45
MathSciNet review: 1039252
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Abstract: Let $ \mathcal{H}$ be the set of all sequences $ h = ({h_k})_{k = 1}^\infty $ of 0s and 1s. A sequence $ x$ in a topological sequence space $ E$ has the property of absolute boundedness $ \vert AB\vert$ if $ \mathcal{H}\cdot x = \{ y\vert{y_k} = {h_k}{x_k},h \in \mathcal{H}\} $ is a bounded subset of $ E$. The subspace $ {E_{\left\vert {AB} \right\vert}}$ of all sequences with absolute boundedness in $ E$ has a natural topology stronger than that induced by $ E$. A sequence $ x$ has the property of absolute sectional convergence $ \vert AK\vert$ if, under this stronger topology, the net $ \{ h\cdot x\} $ converges to $ x$, where $ h$ ranges over all sequences in $ \mathcal{H}$ with a finite number of 1s ordered coordinatewise $ (h' \leq h''\;{\text{iff}}\;\forall k,{h'_k} \leq {h''_k})$. Absolute boundedness and absolute convergence are investigated. It is shown that, for an $ FK$-space $ E$, we have $ E = {E_{\vert AB\vert}}$ if and only if $ E = {l^\infty }\cdot E$, and every element of $ E$ has the property $ \vert AK\vert$ if and only if $ E = {c_0}\cdot E$. Solid hulls and largest solid subspaces of sequence spaces are also considered. The results are applied to standard sequence spaces, convergence fields of matrix methods, classical Banach spaces of Fourier series and to more recently introduced spaces of absolutely and strongly convergent Fourier series.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1039252-0
PII: S 0002-9939(1991)1039252-0
Article copyright: © Copyright 1991 American Mathematical Society