Completion of Countably Seminormed Spaces

Let E be a vector space with a topology generated by countably many increasing seminorms p1 p2 ...and let _{p i} be the completion of E/ker (pi) with respect to pi. If pi are norms compatible in a certain sense, it is known that E is a complete space if and only if E=_{i=1 p i} . In this paper we give a similar characterization of complete spaces in a general case when pi are seminorms, without any additional assumptions. Our characterization coincides with the known one if pi are compatible norms.