  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.
