Riemann Integral of Functions from R into Real Normed Space

Abstract

Summary.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. PreliminariesLet 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 = len D, and (ii) for every natural number i such that i ∈ dom D there exists a point c of X such that c ∈ rng(f divset(D, i)) and it(i) = vol(divset (D, i)) · c. 17

20. Riemann-Stieltjes-Integral

“…The theorem is a consequence of (7). Let us consider a non empty, closed interval subset A of R, a function from A into R, and partial functions u, v, w from R to R. Now we state the propositions:…”

“…Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].…”

Differential Equations on Functions from R into Real Banach Space

“…Now we state the propositions: Then y = g. The theorem is a consequence of (17), (16), (19), and (20).…”

“…The notation and terminology used in this paper have been introduced in the following articles: [29], [5], [11], [3], [6], [7], [19], [13], [34], [31], [33], [1], [15], [25], [32], [18], [24], [23], [26], [27], [20], [2], [8], [14], [16], [28], [12], [37], [38], [9], [35], [36], [17], and [10].…”

Riemann Integral of Functions from R into n-dimensional Real Normed Space

“…The terminology and notation used in this paper have been introduced in the following papers: [23], [24], [6], [2], [25], [8], [7], [1], [4], [3], [5], [20], [10], [14], [12], [13], [18], [22], [19], [26], [9], [11], [15], [17], and [16].…”