## Real Analysis Exchange

### Spaces Of p-Tensor Integrable Functions and Related Banach Space Properties

#### Abstract

In [9] G. F. Stefansson has studied the Banach space $L_1(\nu, X, Y)$, the space of all tensor integrable functions $f : \Omega \to X$ with respect to a countably additive vector valued measure $\nu : \to \Sigma \to Y$ and also the tensor integral of weakly $\nu$-measurable functions. In [1] we obtained some Banach space properties of $L_1(\nu, X, Y)$ and also of w-$L_1(\nu, X, Y)$, the space of all weakly tensor integrable functions. In the present paper, for $1 < p < \infty$, we define the spaces $L_p(\nu, X, Y)$ and w-$L_p(\nu, X, Y)$ of all $\check \otimes_p$-integrable functions and weakly $\check \otimes_p$-integrable functions respectively and discuss several basic properties of these spaces. We also study vector measure duality in $L_p(\nu, X, Y)$ for $1 < p < \infty$.

#### Article information

Source
Real Anal. Exchange, Volume 34, Number 1 (2008), 87-104.

Dates
First available in Project Euclid: 19 May 2009

https://projecteuclid.org/euclid.rae/1242738922

Mathematical Reviews number (MathSciNet)
MR2527124

#### Citation

Chakraborty, N. D.; Basu, Santwana. Spaces Of p -Tensor Integrable Functions and Related Banach Space Properties. Real Anal. Exchange 34 (2008), no. 1, 87--104. https://projecteuclid.org/euclid.rae/1242738922

#### References

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• E. A. Sánchez-Pérez, Vector measure duality and tensor product representations of $L_p$-spaces of vector measures, Proc. Amer. Math. Soc., 132 (2004), 3319–3326.
• G. F. Stefánsson, Integration in vector spaces, Illinois J. Math., 45 (2001), 925–938.
• G. F. Stefánsson, $L_1$ of a vector measure, Matematiche (Catania), 48 (1994), 219–234.
• N. D. Chakraborty and Santwana Basu, On some properties of the space of tensor integrable functions, Anal. Math., 33 (2007), 1–16.
• Guillermo P. Curbera, Operators into $L^1$ of a vector measure and applications to Banach lattices, Math. Ann., 293 (1992), 317–330.
• J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys Monogr., 15 (1977).
• A. Fernández, F. Mayoral, F. Naranjo, C. Sáez and E. A. Sánchez-Pérez, Spaces of $p$-integrable functions with respect to a vector measure, Positivity, 10 (2006), 1–16.
• D. R. Lewis, Integration with respect to vector measures, Pacific J. Math., 33 (1970), 157–165.
• J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, II, Function spaces, Ergeb. Math. Grenzgeb., 97 (1979).
• E. A. Sánchez-Pérez, Compactness arguments for spaces of $p$-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces, Illinois J. Math., 45 (2001), 907–923.
• E. A. Sánchez-Pérez, Vector measure duality and tensor product representations of $L_p$-spaces of vector measures, Proc. Amer. Math. Soc., 132 (2004), 3319–3326.
• G. F. Stefánsson, Integration in vector spaces, Illinois J. Math., 45 (2001), 925–938.
• G. F. Stefánsson, $L_1$ of a vector measure, Matematiche (Catania), 48 (1994), 219–234.