scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary 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 [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

scholarly journals Riemann Integral of Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (2) ◽  
pp. 145-152
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Riemann Integral ◽  
Real Normed Space ◽  
Finite Sequences ◽  
Convergence Of Sequences

Summary In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers. In addition we proved some theorems about the convergence of sequences. We applied definitions introduced in the previous article [21] to the proof of integrability.

scholarly journals Averaging distances in certain Banach spaces

1997 ◽  
Vol 55 (1) ◽  
pp. 147-160 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Finite Dimension ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Empty Interval ◽  
Positive Real ◽  
Complete Discussion ◽  
Image Position

Let E be a Banach space. The averaging interval AI(E) is defined as the set of positive real numbers α, with the following property: For each n ∈ ℕ and for all (not necessarily distinct) x1, x2, … xn ∈ E with ∥x1∥ = ∥x2∥ = … = ∥xn∥ = 1, there is an x ∈ E, ∥x∥ = 1, such thatIt follows immediately, that AI(E) is a (perhaps empty) interval included in the closed interval [1,2]. For example in this paper it is shown that AI(E) = {α} for some 1 < α < 2, if E has finite dimension. Furthermore a complete discussion of AI(C(X)) is given, where C(X) denotes the Banach space of real valued continuous functions on a compact Hausdorff space X. Also a Banach space E is found, such that AI(E) = [1,2].

scholarly journals F. Riesz Theorem

2017 ◽  
Vol 25 (3) ◽  
pp. 179-184
Author(s):  
Keiko Narita ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Closed Interval ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Normed Spaces ◽  
Real Numbers ◽  
Riesz Theorem ◽  
Spaces Of Continuous Functions ◽  
Bounded Functions ◽  
System 1 ◽  
Article 5

Summary In this article, we formalize in the Mizar system [1, 4] the F. Riesz theorem. In the first section, we defined Mizar functor ClstoCmp, compact topological spaces as closed interval subset of real numbers. Then using the former definition and referring to the article [10] and the article [5], we defined the normed spaces of continuous functions on closed interval subset of real numbers, and defined the normed spaces of bounded functions on closed interval subset of real numbers. We also proved some related properties. In Sec.2, we proved some lemmas for the proof of F. Riesz theorem. In Sec.3, we proved F. Riesz theorem, about the dual space of the space of continuous functions on closed interval subset of real numbers, finally. We applied Hahn-Banach theorem (36) in [7], to the proof of the last theorem. For the description of theorems of this section, we also referred to the article [8] and the article [6]. These formalizations are based on [2], [3], [9], and [11].

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals Riemann Integral: Significance and Limitations

2018 ◽  
Vol 7 ◽  
pp. 31-34
Author(s):  
Laxman Bahadur Kunwar
Keyword(s):  
Closed Interval ◽  
Mathematical Science ◽  
Real Analysis ◽  
Lebesgue Integral ◽  
Riemann Integral ◽  
Definition Of

This article basically defines the Riemann integral starting with the definition of partition of a closed interval. It throws a light on the importance and necessity of the Riemann condition of integralbility of a function and explains how the concept of R- integral only on the basis of upper and lower integral is not always practical. It has been suggested that a better rout is to abandon the Reimann integral for Lebesgue integral in real analysis and other fields of mathematical science.

Types over c(k) spaces

2004 ◽  
Vol 77 (1) ◽  
pp. 17-28
Author(s):  
Markus Pomper
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Bounded Functions ◽  
Isolated Point

AbstractLet K be a compact Hausdorff space and C(K) the Banach space of all real-valued continuous functions on K, with the sup norm. Types over C(K) (in the sense of Krivine and Maurey) are represented here by pairs (l, u) of bounded real-valued functions on K, where l is lower semicontinuous and u is upper semicontinuous, l ≤ u and l(x) = u(x) for every isolated point x of K. For each pair the corresponding type is defined by the equation τ(g) = max{║l + g║∞, ║u + g║∞} for all g ∈ C(K), where ║·║∞ is the sup norm on bounded functions. The correspondence between types and pairs (l, u) is bijective.

scholarly journals A note on approximation of continuous functions on normed spaces

2020 ◽  
Vol 12 (1) ◽  
pp. 107-110
Author(s):  
M.A. Mytrofanov ◽  
A.V. Ravsky
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Normed Space ◽  
Analytic Functions ◽  
Real Banach Space ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Normed Spaces ◽  
Approximation Of Continuous Functions

Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions, which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from a complex separable normed space, admitting a separating $*$-polynomial, can be uniformly approximated by $*$-analytic functions.

scholarly journals Differential Equations on Functions from R into Real Banach Space

2013 ◽  
Vol 21 (4) ◽  
pp. 261-272
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Normed Space ◽  
Real Banach Space ◽  
Riemann Integral ◽  
Differentiable Functions ◽  
Real Normed Space

Abstract In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations on real Banach space. We referred to the methods of proof in [30].

scholarly journals SIFAT-SIFAT DASAR INTEGRAL HENSTOCK

2012 ◽  
Vol 6 (2) ◽  
pp. 7-15
Author(s):  
Lexy J. Sinay
Keyword(s):  
Positive Constant ◽  
Positive Function ◽  
Henstock Integral ◽  
Riemann Integral ◽  
Basic Properties ◽  
Definition Of ◽  
Integration Area

This paper was a review about theory of Henstock integral. Riemann gave a definition of integral based on the sum of the partitions in Integration area (interval [a, b]). Thosepartitions is a  -positive constant. Independently, Henstock and Kurzweil replaces - positive constant on construction Riemann integral into a positive function, ie (x)>0 forevery x[a, b]. This function is a partition in interval [a, b]. From this partitions, we can defined a new integral called Henstock integral. Henstock integral is referred to as acomplete Riemann integral, because the basic properties of the Henstock integral is more constructive than Riemann Integral.

scholarly journals Functional Space Consisted by Continuous Functions on Topological Space

2021 ◽  
Vol 29 (1) ◽  
pp. 49-62
Author(s):  
Hiroshi Yamazaki ◽  
Keiichi Miyajima ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Function Space ◽  
Normed Space ◽  
Functional Space ◽  
Continuous Functions ◽  
Bounded Support ◽  
Compact Topological Space ◽  
Definition Of ◽  
System 1

Summary In this article, using the Mizar system [1], [2], first we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space [5]. We prove that this functional space is a Banach space [3]. Next, we give a definition of a function space which is constructed from all continuous functions with bounded support. We also prove that this function space is a normed space.

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