On Closed Subspaces of Grand Lebesgue Spaces

Abstract

We prove a generalized version of a theorem of Grothendieck over finite measure space. We prove a closed subspace of grand Lebesgue space that consist of functions of must be finite dimensional. By using embeddings of Banach spaces and we work inside space . Then we take advantage of many useful properties of Hilbert space.

Keywords

• Grand Lebesgue spaces
• Embeddings
• Hilbert spaces

References

1. T. Iwaniec and C. Sbordone, “On the integrability of Jacobian under minimal hypotheses,” Archive for Rational Mechanics and Analysis, vol. 119, no. 2, pp.129-143,1992.
2. L. Greco, T. Iwaniec, and C. Sbordone, “Inverting the harmonic operator,”manuscripta mathematica, vol.92, pp. 249-258,1997.
3. A. Fiorenza and C. Sbordone, “Existence and uniqueness results for solutions of nonlinear equations with right hand side in , ” Studia Mathematica, vol. 3, no.127, pp. 223-231,1998.
4. R. Castillo and H. Rafeiro, An introductory course in Lebesgue Spaces. Cham, Switzerland: Springer international publishing, 2018.
5. E.M. Stein and R. Shakarchi, Functional analysis: Introduction to further topics in analysis. Princeton, NJ: Princeton University Press, 2011.
6. A. Grothendieck, “Sur certains sous-espaces vectoriels de , ” Canadian Journal of Mathematics, vol. 6, pp.158-160,1954.
7. Y. Kaya, “On a closed subspace of ,” Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol. 9, no. 2, pp. 682-688, 2020.
8. D.V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue spaces: Foundations and Harmonic Analysis. Springer Science & Business Media, 2013.