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On the Euler sequence spaces which include the spaces \(\ell_{p}\) and \(\ell_{\infty}\). I. (English) Zbl 1101.46015

We introduce the Euler sequence space \(e^r_p\) consisting of all sequences whose Euler transforms of order \(r\) are in the space \(\ell_p\) of non-absolute type, which is the BK-space including the space \(\ell_p\) and prove that the spaces \(e^r_p\) and \(\ell_p\), are linearly isomorphic for \(1\leq p\leq \infty\). Furthermore, we give some inclusion relations concerning the space \(e^r_p\). Finally, we determine the \(\alpha\)-, \(\beta\)- and \(\gamma\)-duals of the space \(e^r_p\) for \(1\leq p \leq \infty\) and construct the basis for the space \(e^r_p\), where \(1 \leq p < \infty\).

MSC:

46B45 Banach sequence spaces
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