Real Analysis Exchange

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

Santwana Basu and N. D. Chakraborty

Full-text: Open access

PDF File (218 KB)

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

Permanent link to this document
https://projecteuclid.org/euclid.rae/1242738922

Mathematical Reviews number (MathSciNet)
MR2527124

Subjects
Primary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 28B05: Vector-valued set functions, measures and integrals [See also 46G10]
Secondary: 46B99: None of the above, but in this section

Keywords
Banach space tensor integrable vector measure duality

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


Export citation

References

  • N. D. Chakraborty and Santwana Basu, On some properties of the space of tensor integrable functions, Anal. Math., 33 (2007), 1–16.
    Mathematical Reviews (MathSciNet): MR2312937
    Zentralblatt MATH: 1136.46032
    Digital Object Identifier: doi:10.1007/s10474-007-0101-6
  • Guillermo P. Curbera, Operators into $L^1$ of a vector measure and applications to Banach lattices, Math. Ann., 293 (1992), 317–330.
    Mathematical Reviews (MathSciNet): MR1166123
    Zentralblatt MATH: 0735.46033
    Digital Object Identifier: doi:10.1007/BF01444717
  • J. Diestel and J. J. Uhl, Jr., Vector measures, Mathematical Surveys, 15 American Mathematical Society, Providence, R.I., 1977.
    Mathematical Reviews (MathSciNet): MR453964
  • 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.
    Mathematical Reviews (MathSciNet): MR2223581
    Digital Object Identifier: doi:10.1007/s11117-005-0016-z
  • D. R. Lewis, Integration with respect to vector measures, Pacific J. Math., 33 (1970) 157–165.
    Mathematical Reviews (MathSciNet): MR259064
    Zentralblatt MATH: 0195.14303
    Project Euclid: euclid.pjm/1102977248
  • J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, II, Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, Berlin-New York, 1979.
    Mathematical Reviews (MathSciNet): MR540367
    Zentralblatt MATH: 0403.46022
  • 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.
    Mathematical Reviews (MathSciNet): MR1879243
    Zentralblatt MATH: 0992.46035
    Project Euclid: euclid.ijm/1258138159
  • 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.
    Mathematical Reviews (MathSciNet): MR1879244
    Zentralblatt MATH: 0993.46024
    Project Euclid: euclid.ijm/1258138160
  • G. F. Stefánsson, $L_1$ of a vector measure, Matematiche (Catania), 48 (1994), 219–234.
    Mathematical Reviews (MathSciNet): MR1320665
    Zentralblatt MATH: 0827.46042
  • N. D. Chakraborty and Santwana Basu, On some properties of the space of tensor integrable functions, Anal. Math., 33 (2007), 1–16.
    Mathematical Reviews (MathSciNet): MR2312937
    Zentralblatt MATH: 1136.46032
    Digital Object Identifier: doi:10.1007/s10474-007-0101-6
  • Guillermo P. Curbera, Operators into $L^1$ of a vector measure and applications to Banach lattices, Math. Ann., 293 (1992), 317–330.
    Mathematical Reviews (MathSciNet): MR1166123
    Zentralblatt MATH: 0735.46033
    Digital Object Identifier: doi:10.1007/BF01444717
  • J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys Monogr., 15 (1977).
    Mathematical Reviews (MathSciNet): MR453964
  • 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.
    Mathematical Reviews (MathSciNet): MR2223581
    Digital Object Identifier: doi:10.1007/s11117-005-0016-z
  • D. R. Lewis, Integration with respect to vector measures, Pacific J. Math., 33 (1970), 157–165.
    Mathematical Reviews (MathSciNet): MR259064
    Zentralblatt MATH: 0195.14303
    Project Euclid: euclid.pjm/1102977248
  • J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, II, Function spaces, Ergeb. Math. Grenzgeb., 97 (1979).
    Mathematical Reviews (MathSciNet): MR540367
    Zentralblatt MATH: 0403.46022
  • 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.
    Mathematical Reviews (MathSciNet): MR1879243
    Zentralblatt MATH: 0992.46035
    Project Euclid: euclid.ijm/1258138159
  • 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.
    Mathematical Reviews (MathSciNet): MR1879244
    Zentralblatt MATH: 0993.46024
    Project Euclid: euclid.ijm/1258138160
  • G. F. Stefánsson, $L_1$ of a vector measure, Matematiche (Catania), 48 (1994), 219–234.
    Mathematical Reviews (MathSciNet): MR1320665
    Zentralblatt MATH: 0827.46042

Michigan State University Press

Real Analysis Exchange

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more