A generalization of the Lax equivalence theorem

This paper is concerned with a generalization of the Lax Equivalence Theorem dealing with the approximation of a Co-semigroup in a Hausdorff, locally convex topological vector space, e.g. spaces of tempered distributions. Such spaces are of interest in the study of Cauchy problems for systems of partial differential equations with constant coefficients. Indeed, the solution of a Cauchy problem is generated by the action of a Co-semigroup of transformations in a space of tempered distributions if and only if the Cauchy problem is regular in the sense of Petrowski, cf. [2]. Let V be any locally convex topological vector space over the complex numbers. Following Yosida, [3], we have the following: DEFINITION 1. If { Tt t >01 is a family of bounded linear transformations of V into itself such that (1) TtTt=Tt+sy t, s>O, (2) To=I, and (3) limt 8Ttv = TP for each s>O and each v (E V, then { Tt} is said to be a Co-semigroup. Let {Bh h>O} be a one-parameter family of bounded linear operators in V. DEFINITION 2. The family { Bh } is said to be consistent with { Tt if and only if (Bh Th) Tt/h->O as h->O uniformly with respect to t for every compact interval of time. DEFINITION 3. The family { Bh } is said to be convergent to { Tt } if and only if {Bh-Tt}--*O as nh->t, h->O.