scholarly journals Testing on null sequences is enough for Bochner integrability

1996 ◽  
Vol 54 (2) ◽  
pp. 299-307
Author(s):  
Fernando Mayoral ◽  
Pedro J. Paúl
Keyword(s):  
Uniform Convergence ◽  
Normed Space ◽  
Radon Measure ◽  
Measure Space ◽  
Fréchet Space ◽  
Frechet Space ◽  
Density Condition ◽  
Bochner Integrability

Let E be a normed space, a Fréchet space or a complete (DF)-space satisfying the dual density condition. Let Ω be a Radon measure space. We prove that a function f: Ω → Eis Bochner p-integrable if (and only if) fis p-integrable with respect to the topology of uniform convergence on the norm-null sequences from E′.

On the existence of fixed points in completely continuous operators in F -space

2019 ◽  
pp. 26-32
Author(s):  
Alexander Dorokhov ◽  
Michael Karpov
Keyword(s):  
Fixed Points ◽  
Normed Space ◽  
Fréchet Space ◽  
Frechet Space ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Continuous Operators

This work is dedicated to the development of the theory of fixed points of completely continuous operators. We prove existence of new theorems of fixed points of completely continuous operators in F -space (Frechet space). This class of spaces except Banach includes such important space as a countably normed space and Lp(0 < p < 1), lp(0 < p < 1).

scholarly journals Projective descriptions of the (LF)-spaces of type LB(λp(A), F)

2003 ◽  
Vol 75 (2) ◽  
pp. 163-180
Author(s):  
Angela A. Albanese
Keyword(s):  
Fréchet Space ◽  
Frechet Space ◽  
Density Condition ◽  
Positive Weights

AbstractLet 1 < p < + ∞ or p = 0 and let A = (an)n be an increasing sequence of strictly positive weights on I. Let F be a Fréchet space. It is proved that if λp(A) satisfies the density condition of Heinrich and a certain condition (Ct) holds, then the (LF)-space LBi(λp(A), F) is a topological subspace of Lb(λp(A), F). It is also proved that these conditions are necessary provided F = λq(A) or F contains a complemented copy of Iq with 1 < p ≤ q < +∞.

scholarly journals Distinguishedness of weighted Fréchet spaces of continuous functions

1992 ◽  
Vol 35 (2) ◽  
pp. 271-283 ◽  
Author(s):  
Françoise Bastin
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Density Condition ◽  
Locally Compact ◽  
Spaces Of Continuous Functions ◽  
Locally Compact Hausdorff Space

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

Simultaneous Automorphisms in Spaces of Analytic Functions

1963 ◽  
Vol 15 ◽  
pp. 495-502
Author(s):  
A. Alexiewicz ◽  
M. G. Arsove
Keyword(s):  
Uniform Convergence ◽  
Analytic Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Open Disk ◽  
Compact Sets ◽  
Spaces Of Analytic Functions

Two spaces of analytic functions are considered, each comprised of functions analytic on the open disk NR(0) of radius R (0 < R < +∞ ) centred at the origin. The first space consists of all analytic functions on NR(0) topologized according to the metric of uniform convergence on compact sets. As the second space we allow any Fréchet space of analytic functions on NR(0) for which the topology is stronger than that induced by . Our objective is then to present a scheme for constructing simultaneous automorphisms on and .

scholarly journals Weakly compactly generated Frechet spaces

1979 ◽  
Vol 2 (4) ◽  
pp. 721-724 ◽  
Author(s):  
Surjit Singh Khurana
Keyword(s):  
Measurable Function ◽  
Weak Topology ◽  
Measure Space ◽  
Finite Measure ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Weakly Compact ◽  
Finite Measure Space ◽  
Compactly Generated

It is proved that a weakly compact generated Frechet space is Lindelöf in the weak topology. As a corollary it is proved that for a finite measure space every weakly measurable function into a weakly compactly generated Frechet space is weakly equivalent to a strongly measurable function.

scholarly journals On the projective tensor product of Fréchet spaces

1991 ◽  
Vol 34 (2) ◽  
pp. 169-178 ◽  
Author(s):  
Juan C. Díaz ◽  
Juan A. López Molina
Keyword(s):  
Tensor Product ◽  
Sequence Space ◽  
Positive Answer ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Density Condition ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Montel Space

We are concerned with the following problem. Let F be a Fréchet Montel space and let E be a Fréchet space with a certain property (P). When does it follow that the complete projective tensor product has the property (P)? (We consider the following properties: being Montel, reflexive, satisfying the density condition.) In this paper we provide a positive answer if F is a Montel generalized Dubinsky sequence space with decreasing steps.

Category Results for Tsuji Functions

1977 ◽  
Vol 29 (3) ◽  
pp. 552-558
Author(s):  
D. D. Bonar ◽  
F. W. Carroll ◽  
Peter Colwell
Keyword(s):  
Uniform Convergence ◽  
Unit Disk ◽  
Holomorphic Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Image Position ◽  
Compact Subsets

Let D be the unit disk, |z| < 1, and H(D) the Fréchet space of holomorphic functions on D, provided with the topology of uniform convergence on compact subsets of D. If f is meromorphic in D, we denote by

Darboux chart on projective limit of weak symplectic Banach manifold

2015 ◽  
Vol 12 (07) ◽  
pp. 1550072 ◽  
Author(s):  
Pradip Mishra
Keyword(s):  
Banach Space ◽  
Symplectic Structure ◽  
Reflexive Banach Space ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Banach Manifold ◽  
Boundedness Condition ◽  
Projective System ◽  
Banach Manifolds

Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.

scholarly journals Characterizations and properties of graphs of Baire functions

2018 ◽  
Vol 68 (4) ◽  
pp. 789-802
Author(s):  
Balázs Maga
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Set ◽  
Vertical Section ◽  
Fréchet Space ◽  
Frechet Space ◽  
Similar Question ◽  
Baire Functions

Abstract Let X be a paracompact topological space and Y be a Banach space. In this paper, we will characterize the Baire-1 functions f : X → Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr(f) is the intersection of a sequence $\begin{array}{} \displaystyle (G_n)_{n=1}^{\infty} \end{array}$ of open sets in X × Y such that for all x ∈ X and n ∈ ℕ the vertical section of Gn is a convex set, whose diameter tends to 0 as n → ∞. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.

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