# Semivariation in ${L}^{p}$-spaces

Commentationes Mathematicae Universitatis Carolinae (2005)

- Volume: 46, Issue: 3, page 425-436
- ISSN: 0010-2628

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topJefferies, Brian, and Okada, Susumu. "Semivariation in $L^p$-spaces." Commentationes Mathematicae Universitatis Carolinae 46.3 (2005): 425-436. <http://eudml.org/doc/249537>.

@article{Jefferies2005,

abstract = {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.},

author = {Jefferies, Brian, Okada, Susumu},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {absolutely $p$-summing; bilinear integration; semivariation; tensor product; absolutely -summing; bilinear integration; semivariation; tensor product},

language = {eng},

number = {3},

pages = {425-436},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Semivariation in $L^p$-spaces},

url = {http://eudml.org/doc/249537},

volume = {46},

year = {2005},

}

TY - JOUR

AU - Jefferies, Brian

AU - Okada, Susumu

TI - Semivariation in $L^p$-spaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2005

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 46

IS - 3

SP - 425

EP - 436

AB - 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.

LA - eng

KW - absolutely $p$-summing; bilinear integration; semivariation; tensor product; absolutely -summing; bilinear integration; semivariation; tensor product

UR - http://eudml.org/doc/249537

ER -

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