| Title:
|
Isomorphic and isometric copies of $\ell_\infty(\Gamma)$ in duals of Banach spaces and Banach lattices (English) |
| Author:
|
Wójtowicz, Marek |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
467-471 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^*$ contains an isometric copy of $c_0$ iff $X^*$ contains an isometric copy of $\ell_\infty $, and (2) $E^*$ contains a lattice-isometric copy of $c_0(\Gamma)$ iff $E^*$ contains a lattice-isometric copy of $\ell_\infty(\Gamma)$. (English) |
| Keyword:
|
isometry |
| Keyword:
|
embedding of $\ell_\infty$ |
| Keyword:
|
dual space |
| Keyword:
|
Banach lattice |
| MSC:
|
46B04 |
| MSC:
|
46B25 |
| MSC:
|
46B42 |
| MSC:
|
47B65 |
| idZBL:
|
Zbl 1150.46004 |
| idMR:
|
MR2281008 |
| . |
| Date available:
|
2009-05-05T16:58:44Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119607 |
| . |
| Reference:
|
[1] Abramovich Y.A., Wickstead A.W.: When each continuous operator is regular. II.Indag. Math., N.S. 8 (1997), 281-294. Zbl 0908.47031, MR 1622216 |
| Reference:
|
[2] Aliprantis C.D., Burkinshaw O.: Positive Operators.Academic Press, New York, 1985. Zbl 1098.47001, MR 0809372 |
| Reference:
|
[3] Bessaga C., Pełczyński A.: On basis and unconditional convergence of series in Banach spaces.Studia Math. 17 (1958), 151-164. MR 0115069 |
| Reference:
|
[4] Dowling P.N.: Isometric copies of $c_0$ and $\ell_\infty$ in duals of Banach spaces.J. Math. Anal. Appl. 244 (2000), 223-227. Zbl 0955.46011, MR 1746799 |
| Reference:
|
[5] Lindenstrauss J., Tzafriri L.: Classical Banach Spaces. I.Springer, Berlin, 1977. Zbl 0362.46013, MR 0500056 |
| Reference:
|
[6] Meyer-Nieberg P.: Banach Lattices.Springer, Berlin, 1991. Zbl 0743.46015, MR 1128093 |
| Reference:
|
[7] Rosenthal H.P.: On injective Banach spaces and the spaces $L^\infty(\mu)$ for finite measures $\mu $.Acta Math. 124 (1974), 205-247. MR 0257721 |
| Reference:
|
[8] Rosenthal H.P.: On relatively disjoint families of measures, with some applications to Banach space theory.Studia Math. 37 (1970), 13-36. Zbl 0227.46027, MR 0270122 |
| Reference:
|
[9] Wójtowicz M.: The Sobczyk property and copies of $\ell_\infty$ in locally convex-solid Riesz spaces.Arch. Math. 75 (2000), 376-379. MR 1785446 |
| . |
