Abstract. In this paper we put together some applications given in [7] and [8]. First we deal with the notion of a basis of $\alpha $, $\beta $ type and with the expansion of a function in a power series. Then we deal with the finite section method for operators mapping $s_{\alpha }$ into itself.

AMS Subject Classification (1991): 46A15

Keyword(s): Infinite linear system, operator of first-difference, unital Banach algebra, BK space

Received November 1, 2002, and in final form March 17, 2003. (Registered under 5804/2009.)