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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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

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

First available in Project Euclid: 26 June 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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  • L. Drewnowski, Compact operators on Musielak-Orlicz spaces, Commentationes Mathematicae 27 (1988), 225–232.
  • L. Drewnowski, M. Nawrocki, On the Mackey topology of Orlicz sequence spaces, Arch. Math. 39 (1982), 59–68.
  • N.J. Kalton, Compact and strictly singular operators on Orlicz spaces, Israel J. Math. 26 (1977), no. 2, 126–136.
  • N.J. Kalton, Orlicz sequence spaces without local convexity, Math. Proc. Camb. Phil. Soc. 81 (1977), no. 2, 253–277.
  • N.J. Kalton, N.T. Peck and J.W. Roberts, An $F$-space Sampler, London Math. Society, Lecture Notes series $89$, Cambridge University Press 1984.
  • A. Kamińska and D.M. Kubiak, On the dual of Cesàro function space, Nonlinear Analysis 75 (2012), 2760–2773.
  • A. Kamińska and M. Mastyło, Abstract duality Sawyer's formula and its applications, Monatsh. Math 151 (2007), 223–245.
  • A. Kamińska and Y. Raynaud, New formulas for decreasing rearrangements and a class of Orlicz-Lorentz spaces, Rev. Mat. Complut. (2013), in press.
  • V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press and Nauka, Second Edition, 1982.
  • M. Nawrocki, Fréchet envelopes of locally concave $F$-spaces, Arch. Math. 51 (1988), 363–370.
  • M. Nawrocki and A. Ortyński, The Mackey topology and complemented subspaces of Lorentz sequence spaces $d(w,p)$ for $0<p<1$, Trans. Amer. Math. Soc. 287 (1985), 713–722.
  • A. Pietsch, About the Banach envelope of $l_{1,\infty}$, Rev. Mat. Complut. 22 (2009), no. 1, 209–226.
  • N. Popa, Basic sequences and subspacs in Lorentz sequence spaces without local convexity, Trans. Amer. Math. Soc. 263 (1981), 431–455.
  • J.H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on Hardy and Bergman spaces, Duke Math. J. 43 (1976), 187–202.