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On the mutually non isomorphic $l_{p}(l_{q})$

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In this note we survey the partial results needed to show the following general theorem: ${l_{p}(l_{q}) : 1 ≤ p,q ≤ +∞}$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.
Twórcy
  • Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
autor
  • Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Bibliografia
  • [1] F. Albiac and J.L. Ansorena, On the mutually non isomorphic $l_{p}(l_{q})$ spaces, II, Math. Nachr. 288 (2015), 5-9. doi: 10.1002/mana.201300161
  • [2] F. Albiac and N.J. Kalton, Topics in Banach space theory (Graduate Texts in Mathematics, 233, Springer, New York, 2006).
  • [3] J. Bourgain, P.G. Casazza, J. Lindenstrauss and L. Tzafriri, Banach spaces with a unique unconditional basis, up to permutation, Mem. Amer. Math. Soc. 54, no. 322 (1985).
  • [4] P. Cembranos and J. Mendoza, Banach spaces of vector-valued functions (Lecture Notes in Mathematics 1676, Springer-Verlag, Berlin, 1997).
  • [5] ,P. Cembranos and J. Mendoza, $l_{∞}(l₁)$ and $l₁(l_{∞})$ are not isomorphic, J. Math. Anal. Appl. 341 (2008), 295-297. doi: 10.1016/j.jmaa.2007.10.027
  • [6] P. Cembranos and J. Mendoza, The Banach spaces $l_{∞}(c₀)$ and $c₀(l_{∞})$ are not isomorphic, J. Math. Anal. Appl. 367 (2010), 361-363. doi: 10.1016/j.jmaa.2010.01.057
  • [7] P. Cembranos and J. Mendoza, On the mutually non isomorphic $l_{p}(l_{q})$ spaces, Math. Nachr. 284 (2011), 2013-2023. doi: 10.1002/mana.201010056
  • [8] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators,Cambridge Studies in Advanced Mathematics, 43 (Cambridge University Press, Cambridge, 1995).
  • [9] W.B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972). Israel J. Math. 13 (1972), 301-310. doi: 10.1007/BF02762804
  • [10] T.E. Khmyleva, On the isomorphism of spaces of bounded continuous functions (Russian. English summary), Investigations on linear operators and the theory of functions, XI. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),113,1981,243-246. Translated in Journal of Soviet Mathematics, 22 (1983), 1860-1862. doi: 10.1007/BF01882590
  • [11] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I (Ergebnisse der Mathematik und ihrer Grenzgebiete vol. 92, Springer-Verlag, 1977).
  • [12] J. Motos, M.J. Planells and C.F. Talavera, On some iterated weighted spaces, J. Math. Anal. Appl. 338 (2008), 162-174. doi: 10.1016/j.jmaa.2007.05.009
  • [13] J. Motos and M.J. Planells, On sequence space representations of Hörmander-Beurling spaces, J. Math. Anal. Appl. 348 (2008), 395-403. doi: 10.1016/j.jmaa.2008.07.031
  • [14] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228.
  • [15] H. Triebel, Interpolation theory, function spaces, differential operators (VEB Deutscher Verlag der Wissenschaften, Berlin and North-Holland Publishing Co., Amsterdam-New York 1978 (First editions). Johann Ambrosius Barth, Heidelberg 1995 (Second edition)).
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Bibliografia
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