Riemann Integral of Functions from R into Real Normed Space

Riemann Integral of Functions from R into Real Normed Space In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. The proof method follows the [16].


Preliminaries
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). 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, let D be a Division of A, and let F be a middle volume of f and D. The functor middle sum(f, F ) yielding a point of X is defined by: 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 T be a division sequence of A. A function from N into (the carrier of X) * is said to be a middle volume sequence of f and T if: (Def. 3) For every element k of N holds it(k) is a middle volume of f and T (k).
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, let T be a division sequence of A, let S be a middle volume sequence of f and T , and let k be an element of N. Then S(k) is a middle volume of f and T (k).
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, let T be a division sequence of A, and let S be a middle volume sequence of f and T . The functor middle sum(f, S) yielding a sequence of X is defined as follows: (Def. 4) For every element i of N holds (middle sum(f, S))(i) = middle sum(f, S(i)).

Definition of Riemann Integral on Functions from R into Real Normed Space
Let X be a real normed space, let A be a closed-interval subset of R, and let f be a function from A into the carrier of X. We say that f is integrable if and only if the condition (Def. 5) is satisfied. (Def. 5) There exists a point I of X such that for every division sequence T of A and for every middle volume sequence S of f and T if δ T is convergent and lim(δ T ) = 0, then middle sum(f, S) is convergent and lim middle sum(f, S) = I. We now state three propositions: (1) Let X be a real normed space and R 1 , R 2 , R 3 be finite sequences of elements of X. If len R 1 = len R 2 and Let X be a real normed space, R 1 , R 2 be finite sequences of elements of X, and a be an element of R. If R 2 = a R 1 , then R 2 = a · R 1 . Let X be a real normed space, let A be a closed-interval subset of R, and let f be a function from A into the carrier of X. Let us assume that f is integrable.
The functor integral f yields a point of X and is defined by the condition (Def. 6).
(Def. 6) Let T be a division sequence of A and S be a middle volume sequence of f and T . If δ T is convergent and lim(δ T ) = 0, then middle sum(f, S) is convergent and lim middle sum(f, S) = integral f.
We now state four propositions: (4) Let X be a real normed space, A be a closed-interval subset of R, r be a real number, and f , h be functions from A into the carrier of X. If h = r f and f is integrable, then h is integrable and integral h = r · integral f.
Let X be a real normed space, A be a closed-interval subset of R, and f , g, h be functions from A into the carrier of X. Suppose h = f + g and f is integrable and g is integrable. Then h is integrable and integral h = integral f + integral g.
(7) Let X be a real normed space, A be a closed-interval subset of R, and f , g, h be functions from A into the carrier of X. Suppose h = f − g and f is integrable and g is integrable. Then h is integrable and integral h = integral f − integral g.
Let X be a real normed space, let A be a closed-interval subset of R, and let f be a partial function from R to the carrier of X. We say that f is integrable on A if and only if: (Def. 7) There exists a function g from A into the carrier of X such that g = f A and g is integrable.
Let X be a real normed space, let A be a closed-interval subset of R, and let f be a partial function from R to the carrier of X. Let us assume that A ⊆ dom f.

The functor
A f (x)dx yields an element of X and is defined as follows: (Def. 8) There exists a function g from A into the carrier of X such that g = f A We now state several propositions: (8) Let A be a closed-interval subset of R, f be a partial function from R to the carrier of X, and g be a function from A into the carrier of X. Suppose f A = g. Then f is integrable on A if and only if g is integrable.
(9) Let A be a closed-interval subset of R, f be a partial function from R to the carrier of X, and g be a function from A into the carrier of X. If (10) Let X, Y be non empty sets, V be a real normed space, g, f be partial functions from X to the carrier of V , and g 1 , f 1 be partial functions from Y to the carrier of V . If g = g 1 and f = f 1 , then g 1 + f 1 = g + f. (11) Let X, Y be non empty sets, V be a real normed space, g, f be partial functions from X to the carrier of V , and g 1 , f 1 be partial functions from Y to the carrier of V . If g = g 1 and f = f 1 , then g 1 − f 1 = g − f. (12) Let r be a real number, X, Y be non empty sets, V be a real normed space, g be a partial function from X to the carrier of V , and g 1 be a partial function from Y to the carrier of V . If g = g 1 , then r g 1 = r g.

Linearity of the Integration Operator
Next we state three propositions: (13) Let r be a real number, A be a closed-interval subset of R, and f be a partial function from R to the carrier of X.
(15) Let A be a closed-interval subset of R and f 1 , f 2 be partial functions from R to the carrier of X. Suppose f 1 is integrable on A and f 2 is integrable on A and A ⊆ dom Let X be a real normed space, let f be a partial function from R to the carrier of X, and let a, b be real numbers. The functor b a f (x)dx yielding an element of X is defined as follows: f (x)dx, otherwise. One can prove the following propositions: (16) Let f be a partial function from R to the carrier of X, A be a closedinterval subset of R, and a, b be real numbers. If A = [a, b], then (17) Let f be a partial function from R to the carrier of X and A be a closedinterval subset of R. If vol(A) = 0 and A ⊆ dom f, then f is integrable on A and A f (x)dx = 0 X .
(18) Let f be a partial function from R to the carrier of X, A be a closedinterval subset of R, and a, b be real numbers. If A = [b, a] and A ⊆ dom f,