The present paper is devoted to spaces of real sequences which are transformed by an infinite matrix corresponding to the Euler summation method into convergent sequences or those tending to zero. For both kinds of sequences and, correspondingly, for both spaces, the authors construct Schauder bases and find the conjugate spaces. The infinite matrices transforming the above spaces into , , etc., are found.
Earlier, similar results were obtained for other summation methods; see [P.–N. Ng and P.–Y. Lee, Commentat. Math. 20, 429–433 (1978; Zbl 0408.46012); E. Malkowsky, Mat. Vesn. 49, No. 2, 187–196 (1997; Zbl 0942.40006)].