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 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 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 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 Riemann Integral of Functions from R into Real Normed Space

2011 ◽  
Vol 19 (1) ◽  
pp. 17-22 ◽  
Author(s):  
Keiichi Miyajima ◽  
Takahiro Kato ◽  
Yasunari Shidama
Keyword(s):  
Normed Space ◽  
Riemann Integral ◽  
Real Normed Space ◽  
Range Of Functions ◽  
Riemann Integration

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].

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

2012 ◽  
Vol 20 (1) ◽  
Author(s):  
Keiichi Miyajima ◽  
Artur Korniłowicz ◽  
Yasunari Shidama
Keyword(s):  
Normed Space ◽  
Riemann Integral ◽  
Real Normed Space

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.

scholarly journals On the monotone simultaneous approximation on [0, 1]

1988 ◽  
Vol 38 (3) ◽  
pp. 401-411 ◽  
Author(s):  
Salem M.A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Closed Interval ◽  
Continuous Functions ◽  
Closed Convex Cone ◽  
Measurable Functions ◽  
Decreasing Functions

Let Ω denote the closed interval [0, 1] and let bA denote the set of all bounded, approximately continuous functions on Ω. Let Q denote the Banach space (sup norm) of quasi-continuous functions on Ω. Let M denote the closed convex cone in Q comprised of non-decreasing functions. Let hp, 1 < p < ∞, denote the best Lp-simultaneaous approximation to the bounded measurable functions f and g by elements of M. It is shown that if f and g are elements of Q, then hp converges unifornily to a best L1-simultaneous approximation of f and g. We also show that if f and g are in bA, then hp is continuous.

scholarly journals Stability of a Quartic Functional Equation in Restricted Domains

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Jaeyoung Chung ◽  
Yu-Min Ju
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Stability Problem ◽  
Measure Zero ◽  
Stability Theorem ◽  
Real Normed Space ◽  
Quartic Functional Equation ◽  
Restricted Domains

LetXbe a real normed space andYa Banach space andf:X→Y. We prove the Ulam-Hyers stability theorem for the quartic functional equationf(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)=0in restricted domains. As a consequence we consider a measure zero stability problem of the above inequality whenf:R→Y.

scholarly journals Topological Properties of Real Normed Space

2014 ◽  
Vol 22 (3) ◽  
pp. 209-223
Author(s):  
Kazuhisa Nakasho ◽  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Normed Space ◽  
Lipschitz Continuity ◽  
Inverse Image ◽  
Linear Functions ◽  
Topological Properties ◽  
Normed Spaces ◽  
Real Normed Space ◽  
Convergence Of Sequences ◽  
Normed Subspace

Summary In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on [19](p.3-41), [2] and [34](p.3-67).

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