## $L_p +L_{\infty }$ and $L_p \cap L_{\infty }$ are not isomorphic for all $1 \leq p < \infty$, $p \neq 2$

HTML articles powered by AMS MathViewer

- by Sergey V. Astashkin and Lech Maligranda PDF
- Proc. Amer. Math. Soc.
**146**(2018), 2181-2194 Request permission

## Abstract:

We prove the result stated in the title. It comes as a consequence of the fact that the space $L_p \cap L_{\infty }$, $1\leq p<\infty$, $p\neq 2$, does not contain a complemented subspace isomorphic to $L_p$. In particular, as a subproduct, we show that $L_p \cap L_{\infty }$ contains a complemented subspace isomorphic to ${\ell }_2$ if and only if $p = 2$.## References

- Fernando Albiac and Nigel J. Kalton,
*Topics in Banach space theory*, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR**2192298** - S. V. Astashkin, K. Leśnik, and L. Maligranda,
*Isomorphic structure of Cesàro and Tandori spaces*, Canad. J. Math. (to appear). Preprint of 33 pages submitted on 10 December 2015 at arXiv:1512.03336 - Colin Bennett and Robert Sharpley,
*Interpolation of operators*, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR**928802** - Jöran Bergh and Jörgen Löfström,
*Interpolation spaces. An introduction*, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR**0482275**, DOI 10.1007/978-3-642-66451-9 - S. J. Dilworth,
*Intersection of Lebesgue spaces $L_1$ and $L_2$*, Proc. Amer. Math. Soc.**103**(1988), no. 4, 1185–1188. MR**955005**, DOI 10.1090/S0002-9939-1988-0955005-3 - S. J. Dilworth,
*A scale of linear spaces related to the $L_p$ scale*, Illinois J. Math.**34**(1990), no. 1, 140–158. MR**1031891**, DOI 10.1215/ijm/1255988499 - A. Grothendieck,
*Sur certains sous-espaces vectoriels de $L^p$*, Canad. J. Math.**6**(1954), 158–160 (French). MR**58867**, DOI 10.4153/cjm-1954-017-x - James Hagler and Charles Stegall,
*Banach spaces whose duals contain complemented subspaces isomorphic to $C[0,1]$*, J. Functional Analysis**13**(1973), 233–251. MR**0350381**, DOI 10.1016/0022-1236(73)90033-5 - W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri,
*Symmetric structures in Banach spaces*, Mem. Amer. Math. Soc.**19**(1979), no. 217, v+298. MR**527010**, DOI 10.1090/memo/0217 - N. J. Kalton,
*Lattice structures on Banach spaces*, Mem. Amer. Math. Soc.**103**(1993), no. 493, vi+92. MR**1145663**, DOI 10.1090/memo/0493 - S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov,
*Interpolation of linear operators*, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR**649411** - Joram Lindenstrauss and Lior Tzafriri,
*Classical Banach spaces. II*, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR**540367**, DOI 10.1007/978-3-662-35347-9 - Lech Maligranda,
*The $K$-functional for symmetric spaces*, Interpolation spaces and allied topics in analysis (Lund, 1983) Lecture Notes in Math., vol. 1070, Springer, Berlin, 1984, pp. 169–182. MR**760482**, DOI 10.1007/BFb0099100 - Lech Maligranda,
*The $K$-functional for $p$-convexifications*, Positivity**17**(2013), no. 3, 707–710. MR**3090688**, DOI 10.1007/s11117-012-0200-x - Yves Raynaud,
*Complemented Hilbertian subspaces in rearrangement invariant function spaces*, Illinois J. Math.**39**(1995), no. 2, 212–250. MR**1316534** - Walter Rudin,
*Functional analysis*, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR**1157815** - C. Stegall,
*Banach spaces whose duals contain $l_{1}(\Gamma )$ with applications to the study of dual $L_{1}(\mu )$ spaces*, Trans. Amer. Math. Soc.**176**(1973), 463–477. MR**315404**, DOI 10.1090/S0002-9947-1973-0315404-3 - S. J. Szarek,
*On the best constants in the Khinchin inequality*, Studia Math.**58**(1976), no. 2, 197–208. MR**430667**, DOI 10.4064/sm-58-2-197-208

## Additional Information

**Sergey V. Astashkin**- Affiliation: Department of Mathematics, Samara National Research University, Moskovskoye shosse 34, 443086, Samara, Russia
- MR Author ID: 197703
- Email: astash56@mail.ru
**Lech Maligranda**- Affiliation: Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
- MR Author ID: 118770
- Email: lech.maligranda@ltu.se
- Received by editor(s): June 23, 2017
- Received by editor(s) in revised form: August 10, 2017
- Published electronically: February 1, 2018
- Additional Notes: The research of the first author was partially supported by the Ministry of Education and Science of the Russian Federation, project 1.470.2016/1.4, and by the RFBR grant 17-01-00138.
- Communicated by: Thomas Schlumprecht
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 2181-2194 - MSC (2010): Primary 46E30, 46B20, 46B42
- DOI: https://doi.org/10.1090/proc/13928
- MathSciNet review: 3767368