Eye of the Beholder
Classical Analysis and ODEs
2014-06-02 v1
Abstract
We show that the real line R viewed as a vector space is of uncountable (algebraic) dimension over the scalar field Q of rational numbers. We then build an operator J which maps {R, Q} onto {R, Q}, is Q-linear and whose graph is scattered all over the place, yet is still continuous in the inner product structures on the domain and range spaces. J is not continuous if the usual norm is used on either the domain or range space. We lose continuity and linearity if the scalar field is completed.
Keywords
Cite
@article{arxiv.1405.7968,
title = {Eye of the Beholder},
author = {Nikolas Aksamit and Don Tucker},
journal= {arXiv preprint arXiv:1405.7968},
year = {2014}
}
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6 Pages