Riemann Integral of Functions from R into Real Normed Space

Let X be a real normed space, let A be a closed-interval subset of R, let f be a function from A into the carrier of X, and let D be a Division of A. A finite sequence of elements of X is said to be a middle volume of f and D if it satisfies the conditions (Def. 1). (Def. 1)(i) len it = lenD, and (ii) for every natural number i such that i ∈ domD there exists a point c of X such that c ∈ rng(f divset(D, i)) and it(i) = vol(divset(D, i)) · c.