| Title:
|
Semivariation in $L^p$-spaces (English) |
| Author:
|
Jefferies, Brian |
| Author:
|
Okada, Susumu |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
425-436 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X\hat\otimes_\tau Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X\hat\otimes_\tau Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck's theorem may be applied. In this paper, we take an index $1 \le p < \infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with $\tau_p$ the associated $L^p$-tensor product topology. An application of Orlicz's lemma shows that not all uniformly bounded $X$-valued functions are integrable in $X\hat\otimes_{\tau_p} Y$ with respect to a $Y$-valued measure in the case $1\le p < 2$. For $2 < p <\infty$, the negative result is equivalent to the fact that not all continuous linear maps from $\ell^1$ to $\ell^p$ are $p$-summing, which follows from a result of S. Kwapien. (English) |
| Keyword:
|
absolutely $p$-summing |
| Keyword:
|
bilinear integration |
| Keyword:
|
semivariation |
| Keyword:
|
tensor product |
| MSC:
|
28B05 |
| MSC:
|
46B42 |
| MSC:
|
46G10 |
| MSC:
|
47B65 |
| idZBL:
|
Zbl 1121.28013 |
| idMR:
|
MR2174522 |
| . |
| Date available:
|
2009-05-05T16:52:14Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119538 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
