Eye of the Beholder

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.