Finite Metric Subsets of Banach Spaces

The central idea in this thesis is the introduction of a new isometric invariant of a Banach space. This is Property AI-I. A Banach space has Property AI-I if whenever a finite metric space almost-isometrically embeds into the space, it isometrically embeds. To study this property we introduce two further properties that can be thought of as finite metric variants of Dvoretzky's Theorem and Krivine's Theorem. We say that a Banach space satisfies the Finite Isometric Dvoretzky Property (FIDP) if it contains every finite subset of \ell₂ isometrically. We say that a Banach space has the Finite Isometric Krivine Property (FIKP) if whenever \ell_p is finitely representable in the space then it contains every subset of \ell_p isometrically. We show that every infinite-dimensional Banach space \emph{nearly} has FIDP and every Banach space nearly has FIKP. We then use convexity arguments to demonstrate that not every Banach space has FIKP, and thus we can exhibit classes of Banach spaces that fail to have Property AI-I. The methods used break down when one attempts to prove that there is a Banach space without FIDP and we conjecture that every infinite-dimensional Banach space has Property FIDP.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2019

Author dc:creator

• Kilbane, James

Advisor dc:contributor.advisor

• Zsak, Andras

• Banach Spaces
• Metric Spaces
• Metric Embeddings

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/1de2ab3d-12c6-4598-89b9-721fa4e12bee/download
• https://creativecommons.org/licenses/by-sa/4.0/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.35589

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/288272