Weak Darboux property and transitivity of linear mappings on topological vector spaces

Abstract

It is shown that every linear mapping ontopological vector spaces always has weak Darboux property, therefore, it is continuous ifand only if it is transitive. For finite-dimensional mapping $f$ with values in Hausdorfftopological vector space the following conditions are equivalent: (i) $f$ iscontinuous; (ii) graph of $f$ is closed; (iii) kernel of $f$ is closed; (iv) $f$ istransition map.

Keywords

• Linear mapping
• Darboux property
• transitive mapping
• closed graph
• closed kernel