Sub-Exact Sequence on Hilbert Space (original) (raw)

The notion of the sub-exact sequence is the generalization of exact sequence in algebra, particularly on a module. A module over a ring R is a generalization of the notion of vector space over a field F. A Hilbert space refers to a special vector space over a field F when we have a complete inner product space. The space is complete if every Cauchy sequence converges. Now, we introduce the sub-exact sequence on a Hilbert space, which can be useful later in statistics. This paper is aimed at investigating the properties of the sub-exact sequence and their ratio to direct summand on a Hilbert space. As the result, we obtain two properties of isometric isomorphism sub-exact sequence on a Hilbert space.