On Representing Compact Subsets of Infinite-Dimensional Banach Spaces by Mappings into Euclidean Space

Date

2021-05

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Abstract

Let X be a compact subset of a Banach space B. This dissertation is about inferring properties of X by mapping X into (finite-dimensional) Euclidean spaces. Using the theories of prevalence and projection constants, we formulate conditions under which a typical nonlinear map from B to R^m will preserve the Hausdorff dimension ofX. We show that if X has finite box-counting dimension and m is chosen appropriately, then a typical nonlinear map from B to R ^m will act injectively on X with a Hölder inverse. We provide novel bounds on the Hölder exponent of the inverse.

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Keywords

Banach, Prevalence, Hausdorff dimension, Holder Inverse

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