Eye of the Beholder
  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. 
Nikolas Aksamit and Don Tucker
math.CA