Functiones et Approximatio Commentarii Mathematici

Banach envelopes of $p$-Banach lattices, $0<p<1$, and Cesàro spaces

Anna Kamińska and Pei-Kee Lin

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Abstract

In this note we characterize Banach envelopes of $p$-Banach lattices, $0<p<1$, such that their positive cones are $1$-concave. In particular we show that the Banach envelope of Cesàro sequence space $\widehat{ces_p(v)}$, $0<p<1$, coincides isometrically with the weighted $\ell_1(w)$ space where $w(n) = ||e_n||_{ces_p(v)}= (\sum_{i=n}^\infty i^{-p} v(i))^{1/p}$ and $e_n$ are the unit vectors. For Cesàro function space $Ces_p(v)$, $0<p<1$, its Banach envelope $\widehat{Ces_p(v)}$ is isometrically equal to $L_1(w)$ with $w(t) = (\int_t^\infty s^{-p} v(s) ds)^{1/p}$, $t\in (0,\infty)$.

Article information

Source
Funct. Approx. Comment. Math., Volume 50, Number 2 (2014), 297-306.

Dates
First available in Project Euclid: 26 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1403811846

Digital Object Identifier
doi:10.7169/facm/2014.50.2.7

Mathematical Reviews number (MathSciNet)
MR3229063

Zentralblatt MATH identifier
1312.46005

Subjects
Primary: 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
Secondary: 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42] 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45] 46B04: Isometric theory of Banach spaces 46B42: Banach lattices [See also 46A40, 46B40] 46B45: Banach sequence spaces [See also 46A45]

Keywords
Banach envelopes Mackey topology $p$-Banach lattices for $0<p<1$ Cesàro function and sequence spaces

Citation

Kamińska, Anna; Lin, Pei-Kee. Banach envelopes of $p$-Banach lattices, $0&lt;p&lt;1$, and Cesàro spaces. Funct. Approx. Comment. Math. 50 (2014), no. 2, 297--306. doi:10.7169/facm/2014.50.2.7. https://projecteuclid.org/euclid.facm/1403811846


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