A Review of the Tsirleson's Space Norm

In most cases, the space of all sequences converging to zero or the space of bounded sequences is always embedded in complete normed linear spaces. This concept, however, was modified by B.S. Tsirelson by constructing reflexive complete normed linear spaces with monotone unconditional Schauder basis without embedded copies of sequences converging to zero or the space of bounded sequence. In this article, a relation with any four non-negative integers has been proved, and this concept is used to prove the triangle inequality of a slightly different Tsirelson’s type of norm in the space of all real sequences with finite support. Furthermore, all properties of the norm have been studied for a different type of norm function in the space of real sequences with finite support.