  We develop new approach for studying the abstract Cauchy problem
$\dot{x}=Ax$, $x(0)=x_0\in D(A)$ for linear operators $A$ defined on a locally
convex space $X$. This approach was firstly introduced in the paper
  "Chernoff and Trotter type product formulas" to study the problem for Banach
spaces.
  In this paper we not only generalize the results of the previous paper to
more general topological spaces but also get new results for Banach spaces. In
particular, we prove the "local" extension of Chernoff-Trotter-Kato type
theorems. Applying this result, we prove Chernoff, Lie-Trotter and Trotter-Kato
theorems for locally convex spaces. Also we find necessary and sufficient
conditions for the validity of the Chernoff and Trotter product formulas.
