scholarly journals Analytic Automorphisms and Transitivity of Analytic Mappings

2020 ◽  
Author(s):  
Zoriana Novosad ◽  
Andriy Zagorodnyuk
Keyword(s):  
Composition Operators ◽  
Jacobian Conjecture ◽  
Vector Spaces ◽  
Symmetric Polynomials ◽  
Natural Analogue ◽  
Analytic Operator ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Polynomial Automorphisms ◽  
Linear And Nonlinear

In this paper we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms which show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator which is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclisity of composition operators.

scholarly journals Analytic Automorphisms and Transitivity of Analytic Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2179
Author(s):  
Zoriana Novosad ◽  
Andriy Zagorodnyuk
Keyword(s):  
Composition Operators ◽  
Jacobian Conjecture ◽  
Vector Spaces ◽  
Symmetric Polynomials ◽  
Natural Analogue ◽  
Analytic Operator ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Polynomial Automorphisms ◽  
Linear And Nonlinear

In this paper, we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms that show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator that is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclicity of composition operators.

scholarly journals ULTRAMETRIC AND NON-LOCALLY CONVEX ANALOGUES OF THE GENERAL CURVE LEMMA OF CONVENIENT DIFFERENTIAL CALCULUS

2008 ◽  
Vol 50 (2) ◽  
pp. 271-288
Author(s):  
HELGE GLÖCKNER
Keyword(s):  
Differential Calculus ◽  
Vector Spaces ◽  
Local Convexity ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
General Curve ◽  
Infinite Dimensional Analysis ◽  
Locally Convex ◽  
Differentiability Properties ◽  
Made In

AbstractThe General Curve Lemma is a tool of Infinite-Dimensional Analysis that enables refined studies of differentiability properties of maps between real locally convex spaces to be made. In this article, we generalize the General Curve Lemma in two ways. First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.

scholarly journals Order preserving functions on ordered topological vector spaces

1999 ◽  
Vol 60 (1) ◽  
pp. 55-65 ◽  
Author(s):  
J.C. Candeal ◽  
E. Induráin ◽  
G.B. Mehta
Keyword(s):  
Banach Spaces ◽  
Vector Spaces ◽  
Reflexive Banach Spaces ◽  
Representation Theorems ◽  
Mackey Topology ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Total Preorder

In this paper we prove the existence of continuous order preserving functions on ordered topological vector spaces in an infinite-dimensional setting. In a certain class of topological vector spaces we prove the existence of topologies for which every continuous total preorder has a continuous order preserving representation and show that the Mackey topology is the finest topology with this property. We also prove similar representation theorems for reflexive Banach spaces and for Banach spaces that may not have a pre-dual.

On measurability of real-valued functions in infinite-dimensional topological vector spaces

2018 ◽  
Vol 25 (2) ◽  
pp. 195-199 ◽  
Author(s):  
Mariam Beriashvili ◽  
Tepper Gill ◽  
Aleks Kirtadze
Keyword(s):  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Infinite Dimensional

Abstract Some related questions concerning the measurability properties of real-valued functions with respect to a certain class of measures are discussed.

Inner structure in real vector spaces

2020 ◽  
Vol 27 (3) ◽  
pp. 361-366 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Enrique Naranjo-Guerra
Keyword(s):  
Convex Sets ◽  
Vector Spaces ◽  
Internal Point ◽  
Real Vector ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Intrinsic Characterization ◽  
The One ◽  
Locally Convex

AbstractInternal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifolds, which was not possible by using internal points. We also characterize infinite dimensional real vector spaces by means of the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincides with the one of internal points.

scholarly journals Linearization of certain uniform homeomorphisms

2007 ◽  
Vol 82 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Anthony Weston
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Infinite Dimensional

AbstractThis article concerns the uniform classification of infinite dimensional real topological vector spaces. We examine a recently isolated linearization procedure for uniform homeomorphisms of the form φ: X →Y, where X is a Banach space with non-trivial type and Y is any topological vector space. For such a uniform homeomorphism φ, we show that Y must be normable and have the same supremal type as X. That Y is normable generalizes theorems of Bessaga and Enflo. This aspect of the theory determines new examples of uniform non-equivalence. That supremal type is a uniform invariant for Banach spaces is essentially due to Ribe. Our linearization approach gives an interesting new proof of Ribe's result.

Analytic Subsets of Products of Infinite-Dimensional Projective Spaces

2003 ◽  
Vol 10 (4) ◽  
pp. 603-606
Author(s):  
E. Ballico
Keyword(s):  
Banach Space ◽  
Projective Spaces ◽  
Vector Spaces ◽  
Finite Codimension ◽  
Analytic Subset ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
A Chain

Abstract Let 𝑉𝑖, 1 ≤ 𝑖 ≤ 𝑠, be complex topological vector spaces with 𝑉1 infinite-dimensional and 𝑌 a closed analytic subset of finite codimension of 𝐏(𝑉1) × . . . × 𝐏(𝑉𝑠). Here we show that 𝑌 is algebraic (at least if each 𝑉𝑖 is a Banach space) and that any two points of 𝑌 may be connected by a chain of 𝑠 + 3 lines contained in 𝑌.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

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