Note on initial topologies on rational vector spaces induced by realvalued linear mappings (Q1061953)

Let X be a vector space over rationals, and given a real functional f on X denote by \(\tau_ f\) the topology on X generated by this functional. Assume now that f and g are real and additive functionals on X. The following theorems have been proved. (i) If there exists a non-void open set \(U\subset {\mathbb{R}}\) such that its complement has a non-void interior and \(f^{-1}(U)\in \tau_ g\), then \(f=cg\) for some \(c\in {\mathbb{R}}\). (ii) If \(h: X\to X\) is an additive function, then h: (X,\(\tau\) \({}_ f)\to (X,\tau_ g)\) is continuous iff \(g\circ h=cf\) for some \(c\in {\mathbb{R}}\).