In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article  and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article  and the article  to the proof. Using the definition of the article , we also proved some theorems on bounded functions.