scholarly journals Any infinite-dimensional Fréchet space homeomorphic with its countable product is topologically a Hilbert space

1974 ◽  
Vol 196 ◽  
pp. 93-93
Author(s):  
Wesley E. Terry
Keyword(s):  
Hilbert Space ◽  
Fréchet Space ◽  
Frechet Space ◽  
Infinite Dimensional ◽  
Countable Product

scholarly journals Differentiability of the evolution map and Mackey continuity

2019 ◽  
Vol 31 (5) ◽  
pp. 1139-1177
Author(s):  
Maximilian Hanusch
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Continuity Property ◽  
Fréchet Space ◽  
Frechet Space ◽  
Regularity Problem ◽  
Infinite Dimensional ◽  
Made In

AbstractWe solve the differentiability problem for the evolution map in Milnor’s infinite-dimensional setting. We first show that the evolution map of each {C^{k}}-semiregular Lie group G (for {k\in\mathbb{N}\sqcup\{\mathrm{lip},\infty\}}) admits a particular kind of sequentially continuity – called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which makes it particular among the continuity conditions traditionally considered. As an application of the introduced notions, we discuss the strong Trotter property in the sequentially and the Mackey continuous context. We furthermore conclude that if the Lie algebra of G is a Fréchet space, then G is {C^{k}}-semiregular (for {k\in\mathbb{N}\sqcup\{\infty\}}) if and only if G is {C^{k}}-regular.

scholarly journals An Extension of Hypercyclicity forN-Linear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Juan Bès ◽  
J. Alberto Conejero
Keyword(s):  
Difference Equations ◽  
Entire Functions ◽  
Linear Operators ◽  
Fréchet Space ◽  
Infinite Dimensional ◽  
Bilinear Operators ◽  
Hypercyclic Vectors ◽  
Countable Product ◽  
Dense Orbits ◽  
Residual Sets

Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit forN-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclicN-linear operators, for eachN≥2. Indeed, the nonnormable spaces of entire functions and the countable product of lines supportN-linear operators with residual sets of hypercyclic vectors, forN=2.

scholarly journals On Baire-hyperplane spaces

1979 ◽  
Vol 22 (3) ◽  
pp. 247-255 ◽  
Author(s):  
M. Valdivia
Keyword(s):  
Inductive Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Infinite Dimensional

In this article we prove that in every infinite dimensional separable Fréchet space there is a dense barrelled subspace which is not the inductive limit of Bairehyperplane spaces.

WHITNEY'S TYPE THEOREMS FOR INFINITE DIMENSIONAL SPACES

2000 ◽  
Vol 03 (01) ◽  
pp. 141-160
Author(s):  
S. A. SHKARIN
Keyword(s):  
Linear Extension ◽  
Linear Subspace ◽  
Extension Operator ◽  
Fréchet Space ◽  
Frechet Space ◽  
Entire Space ◽  
Closed Linear Subspace ◽  
Infinite Dimensional ◽  
Class C ◽  
Linear Extension Operator

It is proved that for any f ∈ Ck(L,ℝ), where k ∈ ℕ and L is a closed linear subspace of a nuclear Frechét space X, the function f can be extended to a function of class Ck-1 defined on the entire space X. It is also proved that for any f ∈ Ck (L, ℝ), where k ∈ℕ∪{∞} and L is a closed linear subspace of a conjugate X of a nuclear Frechét space, the function f can be extended to a function of class Ck defined on the entire space X. In addition, it is proved that under these conditions, the existence of a linear extension operator is equivalent to the complementability of the subspace.

scholarly journals On the Existence of Polynomials with Chaotic Behaviour

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Nilson C. Bernardes ◽  
Alfredo Peris
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Linear Manifold ◽  
Fréchet Space ◽  
Frechet Space ◽  
Infinite Dimensional ◽  
Distributional Chaos ◽  
Positive Degree ◽  
Weakly Mixing ◽  
Locally Convex

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.

scholarly journals The existence of a factorized unbounded operator between Fréchet spaces

2018 ◽  
Vol 13 (01) ◽  
pp. 2050017
Author(s):  
Ersin Kızgut ◽  
Murat Yurdakul
Keyword(s):  
Unbounded Operator ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Linear Map ◽  
Infinite Dimensional ◽  
Locally Convex ◽  
Convex Spaces

For locally convex spaces [Formula: see text] and [Formula: see text], the continuous linear map [Formula: see text] is called bounded if there is a zero neighborhood [Formula: see text] of [Formula: see text] such that [Formula: see text] is bounded in [Formula: see text]. Our main result is that the existence of an unbounded operator [Formula: see text] between Fréchet spaces [Formula: see text] and [Formula: see text] which factors through a third Fréchet space [Formula: see text] ends up with the fact that the triple [Formula: see text] has an infinite dimensional closed common nuclear Köthe subspace, provided that [Formula: see text] has the property [Formula: see text].

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