Eye of the Beholder

TL;DR

This paper demonstrates that the real line as a vector space over rationals has uncountable dimension, constructs a special Q-linear operator with a scattered graph that remains continuous under inner product topology but not under norm completion.

Contribution

It introduces a novel Q-linear operator on the real line with a scattered graph that is continuous in inner product topology but not in norm topology, revealing new insights into vector space structures.

Findings

The real line has uncountable algebraic dimension over Q.

Constructed a Q-linear operator with a scattered graph.

Operator is continuous under inner product topology but not under norm completion.

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.