Home Representation of Extendible Bilinear Forms
Article
Licensed
Unlicensed Requires Authentication

Representation of Extendible Bilinear Forms

  • Adam Bowers EMAIL logo
Published/Copyright: December 9, 2015
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 65 Issue 5

Abstract

We show that extendible bilinear forms can be represented in an integral form. The representation requires the use of bimeasures. We then study some properties of these extendible bilinear forms and see how they are related to the Grothendieck inequality.

Keywords: bilinear functionals; extension properties; bimeasures

References

[1] ALBIAC, F.-KALTON, N. J.: Topics in Banach space theory, Grad. Texts in Math. 233, Springer, New York, 2006.Search in Google Scholar

[2] BLEI, R. C.: Analysis in Integer and Fractional Dimensions (1st ed). Cambridge Stud. Adv. Math. 71, Cambridge University Press, Cambridge, 2001.Search in Google Scholar

[3] BOMBAL, F.-PÉREZ-GARC´IA, D.-VILLANUEVA, I.: A decomposition theorem for polymeasures, J. Math. Anal. Appl. 336 (2007), 1316-1323.10.1016/j.jmaa.2007.03.046Search in Google Scholar

[4] CASTILLO, J. M. F. - GARC ´ IA, R. - DEFANT, A. - P´EREZ-GARC´IA, D. - SU´AREZ, J.: Local complementation and the extension of bilinear mappings, Math. Proc. Cambridge Philos. Soc. 152 (2012), 153-166.10.1017/S0305004111000533Search in Google Scholar

[5] CASTILLO, J. M. F.-GARC´IA, R.-JARAMILLO, J. A.: Extension of bilinear forms on Banach spaces, Proc. Amer. Math. Soc. 129 (2001), 3647-3656 (Electronic).10.1090/S0002-9939-01-05986-XSearch in Google Scholar

[6] DIESTEL, J.-FOURIE, J. H.-SWART, J.: The metric theory of tensor products, Amer. Math. Soc., Providence, RI, 2008.10.1090/mbk/052Search in Google Scholar

[7] DIESTEL, J.-UHL, J., Jr.: Vector Measures (1st ed). Math. Surveys Monogr. 15, Amer. Math. Soc., Providence, RI, 1977.10.1090/surv/015/01Search in Google Scholar

[8] DOBRAKOV, I.: On integration in Banach spaces, IX; X (integration with respect to polymeasures), Czechoslovak Math. J. 38(113) (1988), 589-601; 713-725.10.21136/CMJ.1988.102255Search in Google Scholar

[9] DOBRAKOV, I.: Multilinear integration of bounded scalar valued functions, Math. Slovaca 49 (1999), 295-304.Search in Google Scholar

[10] FRÉCHET, M.: Sur les fonctionnelles bilin´eaires, Trans. Amer. Math. Soc. 16 (1915), 215-234.Search in Google Scholar

[11] HAYDEN, T. L.: The extension of bilinear functionals, Pacific J. Math. 22 (1967), 99-108.10.2140/pjm.1967.22.99Search in Google Scholar

[12] JARCHOW, H.-PALAZUELOS, C.-PÉREZ-GARCÍA, D.-VILLANUEVA, I.: Hahn- Banach extension of multilinear forms and summability, J. Math. Anal. Appl. 336 (2007), 1161-1177.10.1016/j.jmaa.2007.03.057Search in Google Scholar

[13] LAHTI, P.-YLINEN, K.: Dilations of positive operator measures and bimeasures related to quantum mechanics, Math. Slovaca, 54 (2004), 169-189.Search in Google Scholar

[14] LINDENSTRAUSS, J.-PeŁCZYŃSKI, A.: Absolutely summing operators in Lp-spaces and their applications, Studia Math. 29 (1968), 275-326.10.4064/sm-29-3-275-326Search in Google Scholar

[15] RAO, M. M.: Characterizations of harmonizable fields, Nonlinear Anal. 63 (2005), 935-947.10.1016/j.na.2005.01.067Search in Google Scholar

[16] RUDIN, W.: Principles of Mathematical Analysis (3rd ed.), McGraw-Hill, Inc., New York, 1976.Search in Google Scholar

[17] RUDIN, W.: Functional Analysis (2nd ed.), McGraw-Hill Science/Engineering/Math., New York, 1991.Search in Google Scholar

[18] YLINEN, K.: Representations of bimeasures, Studia Math. 104 (1993), 269-278.10.4064/sm-104-3-269-278Search in Google Scholar

Received: 2012-11-13
Accepted: 2013-1-29
Published Online: 2015-12-9
Published in Print: 2015-10-1

Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Finite Mixed Sums wih Harmonic Terms
  2. Packing of ℝ2 by Crosses
  3. On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
  4. Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
  5. Generalized Derivations on Lie Ideals and Power Values on Prime Rings
  6. On Monoids of Injective Partial Cofinite Selfmaps
  7. Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
  8. The Controlled Convergence Theorem for the Gap-Integral
  9. The Solvability of a Nonlocal Boundary Value Problem
  10. Oscillation Criteria for Third Order Differential Equations with Functional Arguments
  11. Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
  12. On Null Lagrangians
  13. Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
  14. Convergence of Series on Large Set of Indices
  15. On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
  16. Representation of Extendible Bilinear Forms
  17. Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
  18. Topological Fundamental Groups and Small Generated Coverings
  19. A Relation between two Kinds of Norms for Martingales
  20. Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
  21. Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
  22. Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
https://doi.org/10.1515/ms-2015-0077
Keywords for this article
bilinear functionals; extension properties; bimeasures

Articles in the same Issue

  1. Finite Mixed Sums wih Harmonic Terms
  2. Packing of ℝ2 by Crosses
  3. On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
  4. Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
  5. Generalized Derivations on Lie Ideals and Power Values on Prime Rings
  6. On Monoids of Injective Partial Cofinite Selfmaps
  7. Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
  8. The Controlled Convergence Theorem for the Gap-Integral
  9. The Solvability of a Nonlocal Boundary Value Problem
  10. Oscillation Criteria for Third Order Differential Equations with Functional Arguments
  11. Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
  12. On Null Lagrangians
  13. Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
  14. Convergence of Series on Large Set of Indices
  15. On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
  16. Representation of Extendible Bilinear Forms
  17. Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
  18. Topological Fundamental Groups and Small Generated Coverings
  19. A Relation between two Kinds of Norms for Martingales
  20. Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
  21. Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
  22. Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0077/html
Scroll to top button