Riemann Integral of Functions from R into n-dimensional 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].

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Differential Equations on Functions from R into Real Banach Space

Formalized Mathematics ◽

10.2478/forma-2013-0028 ◽

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].

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Riemann Integral of Functions from ℝ into Real Banach Space

Formalized Mathematics ◽

10.2478/forma-2013-0016 ◽

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.

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Bidual Spaces and Reflexivity of Real Normed Spaces

Formalized Mathematics ◽

10.2478/forma-2014-0030 ◽

2014 ◽

Vol 22 (4) ◽

pp. 303-311

Author(s):

Keiko Narita ◽

Noboru Endou ◽

Yasunari Shidama

Keyword(s):

Real Number ◽

Normed Space ◽

Uniform Boundedness ◽

Linear Operators ◽

Linear Functionals ◽

Normed Spaces ◽

Boundedness Theorem ◽

Natural Mapping ◽

Real Normed Space ◽

Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

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Aplikasi Pemetaan terkait Semi-Inner Produk pada Ruang Bernorma Real

Quadratic: Journal of Innovation and Technology in Mathematics and Mathematics Education ◽

10.14421/quadratic.2021.011-02 ◽

2021 ◽

Vol 1 (1) ◽

pp. 7-14

Author(s):

Aulia Khifah Futhona ◽

Supama

Keyword(s):

Normed Space ◽

Inner Product ◽

Birkhoff Orthogonality ◽

Real Normed Space ◽

Lower Semi

In this article, we give the properties of mappings associated with the upper semi-inner product , lower semi-inner product and Lumer semi-inner product which generate the norm on a real normed space. Furthermore, we establish applications to the Birkhoff orthogonality and characterization of best approximants.

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Maximum average distance in complex finite dimensional normed spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020748 ◽

2002 ◽

Vol 66 (1) ◽

pp. 125-134

Author(s):

Juan C. García-Vázquez ◽

Rafael Villa

Keyword(s):

Unit Sphere ◽

Metric Space ◽

Normed Space ◽

Average Distance ◽

Complex Case ◽

Normed Spaces ◽

Finite Dimensional ◽

Real Normed Space ◽

Dimensional Normed Space

A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xn ∈ M, there exists x ∈ M such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

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On normed spaces with the Wigner Property

Annals of Functional Analysis ◽

10.1007/s43034-019-00035-y ◽

2019 ◽

Vol 11 (3) ◽

pp. 523-539 ◽

Cited By ~ 2

Author(s):

Ruidong Wang ◽

Dariusz Bugajewski

Keyword(s):

Normed Space ◽

Two Dimensional ◽

Normed Spaces ◽

Strictly Convex ◽

Real Normed Space

AbstractThe aim of this paper is to generalize the Wigner Theorem to real normed spaces. A normed space is said to have the Wigner Property if the Wigner Theorem holds for it. We prove that every two-dimensional real normed space has the Wigner Property. We also study the Wigner Property of real normed spaces of dimension at least three. It is also shown that strictly convex real normed spaces possess the Wigner Property.

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Stability of a Quartic Functional Equation in Restricted Domains

Journal of Function Spaces ◽

10.1155/2016/1804206 ◽

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.

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Characterizations of inner product spaces by orthogonal vectors

Journal of Function Spaces and Applications ◽

10.1155/2006/521425 ◽

2006 ◽

Vol 4 (1) ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Marco Baronti ◽

Emanuele Casini

Keyword(s):

Normed Space ◽

Product Space ◽

Closed Ball ◽

Inner Product Space ◽

Inner Product ◽

Product Spaces ◽

Inner Product Spaces ◽

Real Normed Space

LetXbe a real normed space with unit closed ballB. We prove thatXis an inner product space if and only if it is true that wheneverx,yare points in?Bsuch that the line throughxandysupports22Bthenx?yin the sense of Birkhoff.

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NEW SPARSENESS RESULTS ON GRAPH SPANNERS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195995000088 ◽

1995 ◽

Vol 05 (01n02) ◽

pp. 125-144 ◽

Cited By ~ 52

Author(s):

BARUN CHANDRA ◽

GAUTAM DAS ◽

GIRI NARASIMHAN ◽

JOSE SOARES

Keyword(s):

Polynomial Time ◽

Normed Space ◽

Connected Graph ◽

Weighted Graph ◽

Complete Graphs ◽

Weighted Distance ◽

Positive Edge ◽

Real Normed Space ◽

Graph Spanners ◽

Set Of Points

Let G=(V, E) be an n-vertex connected graph with positive edge weights. A subgraph G′=(V, E′) is a t-spanner of G if for all u, v∈V, the weighted distance between u and v in G′ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight. For an arbitrary positive edge-weighted graph G, for any t>1, and any ∈>0, we show that a t-spanner of G with weight [Formula: see text] can be constructed in polynomial time. We also show that (log2 n)-spanners of weight O(1) · wt(MST) can be constructed. We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge xy is the norm of the [Formula: see text] vector. We show that for these graphs, t-spanners with total weight O(log n) · wt(MST) can be constructed in polynomial time.

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