The subset the set of bounded complex sequences of the linear space of complex sequences is defined by the condition (Def. 1). (Def. 1) Let x be a set. Then x ∈ the set of bounded complex sequences if and only if x ∈ the set of complex sequences and idseq(x) is bounded. Let us note that the set of bounded complex sequences is non empty and the set of bounded complex sequences is linearly closed...

We show that for any energy observable every extreme point of the set quantum states with bounded is a pure state. This allows us to write state in terms continuous convex combination energy. Furthermore, we prove finite can be represented as same discuss examples from information theory.

The subset the set of bounded real sequences of the linear space of real sequences is defined by the condition (Def. 1). (Def. 1) Let x be a set. Then x ∈ the set of bounded real sequences if and only if x ∈ the set of real sequences and idseq(x) is bounded. Let us note that the set of bounded real sequences is non empty and the set of bounded real sequences is linearly closed. One can prove th...

let c be a field of subsets of a set i. also, let     1 i i be a non-decreasing positivesequence of real numbers such that 1, 1 0 1   i  and    11 i i . in this paper we prove thatbv of all the games of -bounded variation on c is a non-separable and norm dual banach space of thespace of simple games on c . we use this fact to establish the existence of a linear mapping t from...