Absolute boundedness and absolute convergence in sequence spaces

Buntinas, Martin; Tanović-Miller, Naza

doi:10.1090/S0002-9939-1991-1039252-0

Absolute boundedness and absolute convergence in sequence spaces
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by Martin Buntinas and Naza Tanović-Miller PDF

Proc. Amer. Math. Soc. 111 (1991), 967-979 Request permission

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 $|AB|$ if $\mathcal {H}\cdot x = \{ y|{y_k} = {h_k}{x_k},h \in \mathcal {H}\}$ is a bounded subset of $E$. The subspace ${E_{\left | {AB} \right |}}$ 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 $|AK|$ 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 $E = {E_{|AB|}}$ if and only if $E = {l^\infty }\cdot E$, and every element of $E$ has the property $|AK|$ 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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