dc.contributor.author Kilbane, James dc.date.accessioned 2019-01-21T16:43:56Z dc.date.available 2019-01-21T16:43:56Z dc.date.issued 2019-01-31 dc.date.submitted 2018-07-20 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/288272 dc.description.abstract 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 $\ell_2$ 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. dc.language.iso en dc.rights Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) dc.rights.uri https://creativecommons.org/licenses/by-sa/4.0/ dc.subject Banach Spaces dc.subject Metric Spaces dc.subject Metric Embeddings dc.title Finite Metric Subsets of Banach Spaces dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.publisher.department DPMMS dc.date.updated 2019-01-05T12:15:02Z dc.identifier.doi 10.17863/CAM.35589 dc.type.qualificationtitle PhD in Pure Maths and Math Statistics cam.supervisor Zsak, Andras cam.thesis.funding false
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