The Grothendieck property of Weak Lp spaces and Marcinkiewicz spaces

Master Thesis (2019)
Author(s)

P. van den Bosch (TU Delft - Electrical Engineering, Mathematics and Computer Science)

Contributor(s)

Ben de Pagter – Mentor (TU Delft - Analysis)

MC Veraar – Graduation committee member (TU Delft - Analysis)

Richard KRAAIJ – Graduation committee member (TU Delft - Applied Probability)

M de Jeu – Graduation committee member (Universiteit Leiden)

Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright
© 2019 Pascal van den Bosch
More Info
expand_more
Publication Year
2019
Language
English
Copyright
© 2019 Pascal van den Bosch
Graduation Date
01-03-2019
Awarding Institution
Delft University of Technology
Programme
Applied Mathematics
Faculty
Electrical Engineering, Mathematics and Computer Science
Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Abstract

In this master thesis the proof of Lotz that Weak Lp spaces have the Grothendieck property is studied. The proof is slightly modified to be more explicit and easier to comprehend by introducing lemma’s to better separate different parts of the proof that more clearly reveal its structure. Furthermore, the more general Marcinkiewicz spaces are shown to sometimes have the Grothendieck property, using the sufficient conditions for a Banach lattice to have the Grothendieck property that Lotz derived to prove the Grothendieck property of Weak Lp spaces. For most of these conditions that together are sufficient, proving that Marcinkiewicz spaces satisfy them is done in a way very similar to the case of Weak Lp spaces. However, the proof of the (necessary) condition that the dual sometimes has order continuous norm does not allow for such a simple generalization and requires more work. Finally, by using some more recent results about the existence of symmetric functionals in the dual, the conditions that are given for the Grothendieck property of Marcinkiewicz spaces are shown to be necessary, and we thereby obtain a characterization of the Marcinkiewicz spaces that have the Grothendieck property

Files

Verslag.pdf
(pdf | 0.524 Mb)
License info not available