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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 locallyconvex space $X$. This approach was firstly introduced in the paper  "Chernoff and Trotter type product formulas" to study the problem for Banachspaces.  In this paper we not only generalize the results of the previous paper tomore general topological spaces but also get new results for Banach spaces. Inparticular, we prove the "local" extension of Chernoff-Trotter-Kato typetheorems. Applying this result, we prove Chernoff, Lie-Trotter and Trotter-Katotheorems for locally convex spaces. Also we find necessary and sufficientconditions for the validity of the Chernoff and Trotter product formulas.

Chernoff and Trotter-Kato theorems for locally convex spaces