scholarly journals Locally convex linear topological spaces that are homeomorphic to the powers of the real line

1991 ◽  
Vol 113 (2) ◽  
pp. 599-599 ◽  
Author(s):  
A. Chigogidze
Keyword(s):  
Real Line ◽  
Topological Spaces ◽  
The Real ◽  
Locally Convex

Linear quasi-differential operators in locally integrable spaces on the real line

2000 ◽  
Vol 130 (4) ◽  
pp. 671-698 ◽  
Author(s):  
R. R. Ashurov ◽  
W. N. Everitt
Keyword(s):  
Banach Spaces ◽  
Real Line ◽  
Differential Operators ◽  
Linear Topological Space ◽  
Topological Spaces ◽  
The Real ◽  
Conjugate Space ◽  
Linear Differential ◽  
Definition Of ◽  
Locally Convex

The theory of ordinary linear quasi-differential expressions and operators has been extensively developed in integrable-square Hilbert spaces. There is also an extensive theory of ordinary linear differential expressions and operators in integrable-p Banach spaces.However, the basic definition of linear quasi-differential expressions involves Lebesgue locally integrable spaces on intervals of the real line. Such spaces are not Banach spaces but can be considered as complete locally convex linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete locally convex linear topological space, but now with the topology derived as a strict inductive limit.This paper develops the properties of linear quasi-differential operators in a locally integrable space and the first conjugate space. Conjugate and preconjugate operators are defined in, respectively, dense and total domains.

scholarly journals Almost-periodicity in linear topological spaces and applications to abstract differential equations

1984 ◽  
Vol 7 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Gaston Mandata N'Guerekata
Keyword(s):  
Continuous Function ◽  
Real Line ◽  
Locally Convex Space ◽  
Topological Spaces ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
Fréchet Spaces ◽  
Abstract Differential Equations ◽  
T Tau ◽  
Locally Convex

LetEbe a complete locally convex space (l.c.s.) andf:R→Ea continuous function; thenfis said to be almost-periodic (a.p.) if, for every neighbourhood (of the origin inE)U, there existsℓ=ℓ(U)>0such that every interval[a,a+ℓ]of the real line contains at least oneτpoint such thatf(t+τ)−f(t)∈Ufor everyt∈R. We prove in this paper many useful properties of a.p. functions in l.c.s, and give Bochner's criteria in Fréchet spaces.

scholarly journals Wadge-like reducibilities on arbitrary quasi-Polish spaces

2014 ◽  
Vol 25 (8) ◽  
pp. 1705-1754 ◽  
Author(s):  
LUCA MOTTO ROS ◽  
PHILIPP SCHLICHT ◽  
VICTOR SELIVANOV
Keyword(s):  
Real Line ◽  
Polish Space ◽  
Baire Space ◽  
Topological Spaces ◽  
Polish Spaces ◽  
The Real ◽  
Degree Structure ◽  
Wadge Hierarchy

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

Countable Compagtifications

1966 ◽  
Vol 18 ◽  
pp. 616-620 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Positive Integer ◽  
Compact Space ◽  
Real Line ◽  
Topological Spaces ◽  
Dense Subspace ◽  
Natural Numbers ◽  
Finite Sets ◽  
The Real ◽  
One To One

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.

(LF)-spaces with absolute bases

1970 ◽  
Vol 67 (2) ◽  
pp. 283-286 ◽  
Author(s):  
G. Bennett ◽  
J. B. Cooper
Keyword(s):  
Complex Plane ◽  
Real Line ◽  
Convex Space ◽  
Inductive Limit ◽  
Locally Convex Space ◽  
The Real ◽  
Continuous Seminorm ◽  
Image Position ◽  
Locally Convex ◽  
Absolute Basis

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely aswhere ξk ∈K for each k. If this representation converges absolutely, i.e. iffor every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.

scholarly journals On a Lindenbaum Composition Theorem

2019 ◽  
Vol 74 (1) ◽  
pp. 145-158
Author(s):  
Jaroslav Šupina ◽  
Dávid Uhrik
Keyword(s):  
Topological Space ◽  
Real Line ◽  
Topological Spaces ◽  
The Real ◽  
Measurable Functions ◽  
Borel Measurable ◽  
Composition Theorem ◽  
Perfectly Normal

Abstract We discuss several questions about Borel measurable functions on a topological space. We show that two Lindenbaum composition theorems [Lindenbaum, A. Sur les superpositions des fonctions représentables analytiquement, Fund. Math. 23 (1934), 15–37] proved for the real line hold in perfectly normal topological space as well. As an application, we extend a characterization of a certain class of topological spaces with hereditary Jayne-Rogers property for perfectly normal topological space. Finally, we pose an interesting question about lower and upper Δ02-measurable functions.

Ideals of porous sets in the real line and in metrizable topological spaces

2000 ◽  
Vol 102 (5) ◽  
pp. 4508-4522 ◽  
Author(s):  
O. L. Semenova ◽  
A. A. Florinskii
Keyword(s):  
Real Line ◽  
Topological Spaces ◽  
The Real ◽  
Porous Sets

scholarly journals THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE

2020 ◽  
Vol 6 (2) ◽  
pp. 108
Author(s):  
Tursun K. Yuldashev ◽  
Farhod G. Mukhamadiev
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Real Line ◽  
Local Density ◽  
Cardinal Number ◽  
Topological Spaces ◽  
Topological Properties ◽  
Locally Compact ◽  
The Real ◽  
Weak Density

In this paper, the local density \((l d)\) and the local weak density \((l w d)\) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree \(S P^{n}\) and the subfunctor of permutation degree \(S P_{G}^{n}\),  \(P\) is the cardinal number of topological spaces. Let \(X\) be an infinite \(T_{1}\)-space. We prove that the following propositions hold.(1) Let \(Y^{n} \subset X^{n}\); (A) if \(d\, \left(Y^{n} \right)=d\, \left(X^{n} \right)\), then \(d\, \left(S P^{n} Y\right)=d\, \left(SP^{n} X\right)\); (B) if \(l w d\, \left(Y^{n} \right)=l w d\, \left(X^{n} \right)\), then \(l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)\). (2) Let \(Y\subset X\); (A) if \(l d \,(Y)=l d \,(X)\), then \(l d\, \left(S P^{n} Y\right)=l d\, \left(S P^{n} X\right)\); (B) if \(w d \,(Y)=w d \,(X)\), then \(w d\, \left(S P^{n} Y\right)=w d\, \left(S P^{n} X\right)\).(3) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is a locally compact \(T_{1}\)-space, then \(S P^{n} X, \, S P_{G}^{n} X\), and \(\exp _{n} X\) are \(k\)-spaces.(4) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is an infinite \(T_{1}\)-space, then \(n \,\pi \,w \left(X\right)=n \, \pi \,w \left(S P^{n} X \right)=n \,\pi \,w \left(S P_{G}^{n} X \right)=n \,\pi \,w \left(\exp _{n} X \right)\).We also have studied that the functors \(SP^{n},\) \(SP_{G}^{n} ,\) and \(\exp _{n} \) preserve any \(k\)-space. The functors \(SP^{2}\) and \(SP_{G}^{3}\) do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite \(T_{1}\)-space \(X\) coincides with the densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\). It is also shown that the weak density of an infinite \(T_{1}\)-space \(X\) coincides with the weak densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\).

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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