The Series on Banach Algebra

Let X be a non empty normed structure and let s1 be a sequence of X. The functor ( ∑ κ α=0(s1)(α))κ∈N yielding a sequence of X is defined as follows: (Def. 1) ( ∑ κ α=0(s1)(α))κ∈N(0) = s1(0) and for every natural number n holds ( ∑ κ α=0(s1)(α))κ∈N(n + 1) = ( ∑ κ α=0(s1)(α))κ∈N(n) + s1(n + 1). One can prove the following proposition (1) Let X be an add-associative right zeroed right complementable non empty normed structure and s1 be a sequence of X. Suppose that for every natural number n holds s1(n) = 0X . Let m be a natural number. Then ( ∑ κ α=0(s1)(α))κ∈N(m) = 0X . Let X be a real normed space and let s1 be a sequence of X. We say that s1 is summable if and only if: (Def. 2) ( ∑ κ α=0(s1)(α))κ∈N is convergent. Let X be a real normed space. One can verify that there exists a sequence of X which is summable. Let X be a real normed space and let s1 be a sequence of X. The functor