Chernoff and Trotter-Kato theorems for locally convex spaces

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.