We develop a rather general version of Feynman's operational calculus. Instead of using operator-valued (or L(X)-valued) functions in the operational calculus, we use functions taking values in a topological algebra E. While this complicates some aspects of the operational calculus, it supplies a quite general framework in which the time-ordering calculations required for the operational calculus can be carried out, allowing the operational calculus to be used with functions taking values in any algebra which satisfies the conditions which are imposed on the topological algebras considered in this paper.

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