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arXiv · 1405.7968

Eye of the Beholder

Nikolas Aksamit·Don Tucker
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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.

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Nikolas Aksamit, Don Tucker. 2014-05-30. Eye of the Beholder. https://arxiv.org/abs/1405.7968

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