  The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram
Theorem where completeness of one of the spaces is not complete. In this paper,
RTL is deduced from an operator-theoretical version on normed space. The main
point of the paper is a theory of derivations, based on RTL, for which
well-posedness is proved. One application concerns non-autonomous evolution
equations with a new initial-value and a periodic boundary condition for the
time variable.
