w * -basic sequences and reflexivity of Banach spaces

Kamil John

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 3, page 677-681
  • ISSN: 0011-4642

Abstract

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We observe that a separable Banach space X is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if ℒ ( X , Y ) is not reflexive for reflexive X and Y then ℒ ( X 1 , Y ) is is not reflexive for some X 1 ⊂ X , X 1 having a basis.

How to cite

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John, Kamil. "$w^*$-basic sequences and reflexivity of Banach spaces." Czechoslovak Mathematical Journal 55.3 (2005): 677-681. <http://eudml.org/doc/30977>.

@article{John2005,
abstract = {We observe that a separable Banach space $X$ is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if $\mathcal \{L\}(X,Y)$ is not reflexive for reflexive $X$ and $Y$ then $\mathcal \{L\}(X_1, Y)$ is is not reflexive for some $X_1\subset X$, $X_1$ having a basis.},
author = {John, Kamil},
journal = {Czechoslovak Mathematical Journal},
keywords = {reflexive Banach space; Schauder basis; quotient space; w$^*$-basic sequence; tensor product; reflexive Banach space; Schauder basis; quotient space; w-basic sequence; tensor product},
language = {eng},
number = {3},
pages = {677-681},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$w^*$-basic sequences and reflexivity of Banach spaces},
url = {http://eudml.org/doc/30977},
volume = {55},
year = {2005},
}

TY - JOUR
AU - John, Kamil
TI - $w^*$-basic sequences and reflexivity of Banach spaces
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 677
EP - 681
AB - We observe that a separable Banach space $X$ is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if $\mathcal {L}(X,Y)$ is not reflexive for reflexive $X$ and $Y$ then $\mathcal {L}(X_1, Y)$ is is not reflexive for some $X_1\subset X$, $X_1$ having a basis.
LA - eng
KW - reflexive Banach space; Schauder basis; quotient space; w$^*$-basic sequence; tensor product; reflexive Banach space; Schauder basis; quotient space; w-basic sequence; tensor product
UR - http://eudml.org/doc/30977
ER -

References

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  1. 10.1090/S0002-9939-1972-0288560-8, Proc. Amer. Math. Soc. 31 (1972), 109–111. (1972) MR0288560DOI10.1090/S0002-9939-1972-0288560-8
  2. Sequences and Series in Banach Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1984. (1984) MR0737004
  3. Functional Analysis and Infinite Dimensional Geometry, Canad. Math. Soc. Books in Mathematics Springer-Verlag, New York, 2001. (2001) MR1831176
  4. On the reflexivity of the Banach space  L ( X , Y ) , Funkts. Anal. Prilozh. 8 (1974), 97–98. (Russian) (1974) MR0342991
  5. Reflexivity of  L ( E , F ) , Proc. Amer. Math. Soc. 39 (1974), 175–177. (1974) MR0315407
  6. Locally Convex Spaces, Teubner-Verlag, Stuttgart, 1981. (1981) Zbl0466.46001MR0632257
  7. 10.4064/sm-43-1-77-92, Studia Math. 43 (1972), 77–92. (1972) MR0310598DOI10.4064/sm-43-1-77-92
  8. Classical Banach Spaces I. Sequence Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete  92, Springer-Verlag, Berlin-Heidelberg-Berlin, 1977. (1977) MR0500056
  9. Reflexive spaces of homogeneous polynomials, Bull. Polish Acad. Sci. Math. 49 (2001), 211–222. (2001) Zbl1068.46027MR1863260
  10. 10.4064/sm-21-3-370-374, Studia Math. 21 (1962), 371–374. (1962) MR0146636DOI10.4064/sm-21-3-370-374
  11. Biorthogonal systems and reflexivity of Banach spaces, Czechoslovak Math.  J. 9 (1959), 319–325. (1959) MR0110008
  12. Reflexivity of L ( E , F ) , Proc. Am. Math. Soc. 34 (1972), 171–174. (1972) Zbl0242.46018MR0291777
  13. 10.4064/sm-21-3-351-369, Studia Math. 21 (1962), 351–369. (1962) Zbl0114.30903MR0146635DOI10.4064/sm-21-3-351-369

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