The Banach Space

Let x be a sequence of real numbers and let p be a real number. The functor x yielding a sequence of real numbers is defined as follows: (Def. 1) For every natural number n holds x(n) = |x(n)|. Let p be a real number. Let us assume that p ≥ 1. The functor l yielding a non empty subset of the carrier of the linear space of real sequences is defined as follows: (Def. 2) For every set x holds x ∈ l iff x ∈ the set of real sequences and (idseq(x)) p is summable. In the sequel a, b, c are real numbers. We now state several propositions: (1) If a ≥ 0 and a < b and c > 0, then a < b. (2) Let p be a real number. Suppose 1 ≤ p. Let a, b be sequences of real numbers and n be a natural number. Then ( ∑κ α=0((a+b) )(α))κ∈N(n) 1