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Let X be an arbitrary vector space (without topology) together with a pair of endomorphisms D , RCLO(X) such that D R = I . If the kernel Z of D is nontrivial, the product space X,: = )(Z has a corresponding pair of endomorphisms Do, Ro E L,(X,,) with DoRo = lo and we can find a homomorphic representation @: X -X o such that @D=DO@ and @R= Ro@ [3]. This certain problems in S, for instance abstract differential equations [I], can be transformed into associated problems in the representation space X, and analysed, taking advantage of the simpler structure of X, as a linear sequence space. In particular, the space X o is endowed with a nontrivial structural compatible topology such that X, is metrizable and Ex,, converges iff (2,) -0.Using an appropriate notion of contractivity

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