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 ( X , Y ) is not reflexive for reflexive X and Y then ( 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. On w *   basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77–92. (1972) MR0310598
  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. A note on the paper of I.  Singer “Basic sequences and reflexivity of Banach spaces”, Studia Math. 21 (1962), 371–374. (1962) MR0146636
  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. Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1962), 351–369. (1962) Zbl0114.30903MR0146635

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