scholarly journals On properly separable quotients of strict (LF) Spaces

1989 ◽  
Vol 47 (2) ◽  
pp. 307-312 ◽  
Author(s):  
W. J. Robertson
Keyword(s):  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Inductive Limit ◽  
General Question ◽  
Fréchet Spaces ◽  
Infinite Dimensional

AbstractAll known Banach spaces have an infinite-dimensional separable quotient and so do all nonnormable Fréchet spaces, although the general question for Banach spaces is still open. A properly separable topological vector space is defined, in such a way that separable and properly separable are equivalent for an infinite-dimensional complete metrisable space. The main result of this paper is that the strict inductive limit of a sequence of non-normable Fréchet spaces has a properly separable quotient.

On the Dimension of a Complete Metrizable Topological Vector Space

1977 ◽  
Vol 20 (2) ◽  
pp. 271-272 ◽  
Author(s):  
J. O. Popoola ◽  
I. Tweddle
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Fréchet Spaces ◽  
Convex Case ◽  
Space Case ◽  
Locally Convex ◽  
Elegant Proof ◽  
Metrizable Topological Vector Space

The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments.

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.

scholarly journals On a Semigroup Problem II

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1392
Author(s):  
Viorel Nitica ◽  
Andrew Torok
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Infinite Dimensional ◽  
Finite Dimensional

We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer is positive, we give a new example of infinite dimensional spaces where the answer is negative.

scholarly journals Completely superharmonic measures for the infinitesimal generator A of a diffusion semi-group and positive eigen elements of A

1981 ◽  
Vol 83 ◽  
pp. 53-106 ◽  
Author(s):  
Masayuki Itô ◽  
Noriaki Suzuki
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Infinitesimal Generator ◽  
Hausdorff Space ◽  
Inductive Limit ◽  
Semi Group ◽  
Radon Measures ◽  
Countable Basis ◽  
Inductive Limit Topology ◽  
Locally Compact Hausdorff Space

Let X be a locally compact Hausdorff space with countable basis. We denote byM(X) the topological vector space of all real Radon measures in X with the vague topology,MK(X) the topological vector space of all real Radon measures in X whose supports are compact with the usual inductive limit topology.

scholarly journals Inductive Limits of Banach Spaces

1976 ◽  
Vol 19 (4) ◽  
pp. 495-496 ◽  
Author(s):  
Steven F. Bellenot
Keyword(s):  
Banach Spaces ◽  
Inductive Limit ◽  
Inductive Limits ◽  
Infinite Dimensional

AbstractIf X and Y are infinite-dimensional Banach spaces, then Y is the inductive limit of Banach spaces each isomorphic to X.

Probability Measures with Big Kernels

2001 ◽  
Vol 8 (2) ◽  
pp. 333-346
Author(s):  
Nguyen Duy Tien ◽  
V. Tarieladze
Keyword(s):  
Probability Measure ◽  
Vector Space ◽  
Topological Vector Space ◽  
Dual Space ◽  
Full Measure ◽  
Probability Measures ◽  
Infinite Dimensional ◽  
Cylindrical Measure ◽  
Second Category

Abstract It is shown that in an infinite-dimensional dually separated second category topological vector space X there does not exist a probability measure μ for which the kernel coincides with X. Moreover, we show that in “good” cases the kernel has the full measure if and only if it is finitedimensional. Also, the problem posed by S. Chevet [Kernel associated with a cylindrical measure, Springer-Verlag, 1981, p. 69] is solved by proving that the annihilator of the kernel of a measure μ coincides with the annihilator of μ if and only if the topology of μ-convergence in the dual space is essentially dually separated.

scholarly journals Pre-Schauder Bases in Topological Vector Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1026 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Francisco Javier Pérez-Fernández
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Linear Span ◽  
Vector Spaces ◽  
Schauder Basis ◽  
Topological Vector Spaces ◽  
Hausdorff Topological Vector Space ◽  
Schauder Bases

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

Representations of the spaces Cm(Ω) ∩ Hk, p (Ω)

1996 ◽  
Vol 120 (3) ◽  
pp. 489-498 ◽  
Author(s):  
A. A. Albanese ◽  
G. Metafune ◽  
V. B. Moscatelli
Keyword(s):  
Banach Spaces ◽  
Function Space ◽  
Inductive Limit ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Complemented Subspaces ◽  
Complemented Subspace ◽  
Strong Dual

The present work has its motivation in the papers [2] and [6] on distinguished Fréchet function spaces. Recall that a Fréchet space E is distinguished if it is the projective limit of a sequence of Banach spaces En such that the strong dual E′β is the inductive limit of the sequence of the duals E′n. Clearly, the property of being distinguished is inherited by complemented subspaces and in [6] Taskinen proved that the Fréchet function space C(R) ∩ L1(R) (intersection topology) is not distinguished, by showing that it contains a complemented subspace of Moscatelli type (see Section 1) that is not distinguished. Because of the criterion in [1], it is easy to decide when a Frechet space of Moscatelli type is distinguished. Using this, in [2], Bonet and Taskinen obtained that the spaces open in RN) are distinguished, by proving that they are isomorphic to complemented subspaces of distinguished Fréchet spaces of Moscatelli type.

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