Egorov measurability and generator cores

Introduction. Let f be a modulus, ei = (δij) and E = {ei, i = 1, 2, …}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp, 0 < p ≦ 1 is an L(f) space with f(x) = xp, and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l1, for all f modulus (see [2]), and l1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space.

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In what spaces is every closed normal cone regular?

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150000938x ◽

1970 ◽

Vol 17 (2) ◽

pp. 121-125 ◽

Cited By ~ 8

Author(s):

C. W. McArthur

Keyword(s):

Banach Space ◽

Convex Space ◽

Unconditional Basis ◽

Normal Cone ◽

Closed Subspace ◽

Locally Convex Space ◽

Regular Part ◽

Fréchet Space ◽

Sequential Completeness ◽

Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

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Completeness of the L1-space of closed vector measures

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028893 ◽

1990 ◽

Vol 33 (1) ◽

pp. 71-78 ◽

Cited By ~ 5

Author(s):

Werner J. Ricker

Keyword(s):

Uniform Convergence ◽

Convex Space ◽

Vector Measure ◽

Sufficient Conditions ◽

Locally Convex Space ◽

Vector Measures ◽

Locally Convex

The notion of a closed vector measure m, due to I. Kluv´;nek, is by now well established. Its importance stems from the fact that if the locally convex space X in which m assumes its values is sequentially complete, then m is closed if and only if its L1-space is complete for the topology of uniform convergence of indefinite integrals. However, there are important examples of X-valued measures where X is not sequentially complete. Sufficient conditions guaranteeing the completeness of L1(m) for closed X-valued measures m are presented without the requirement that X be sequentially complete.

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Nuclear operators on Banach function spaces

Positivity ◽

10.1007/s11117-020-00787-1 ◽

2020 ◽

Author(s):

Marian Nowak

Keyword(s):

Banach Space ◽

Convex Space ◽

Banach Function Space ◽

Finite Measure ◽

Locally Convex Space ◽

Nuclear Operator ◽

Representing Measure ◽

Nuclear Operators ◽

Nikodym Property ◽

Locally Convex

Abstract Let X be a Banach space and E be a perfect Banach function space over a finite measure space $$(\Omega ,\Sigma ,\lambda )$$ ( Ω , Σ , λ ) such that $$L^\infty \subset E\subset L^1$$ L ∞ ⊂ E ⊂ L 1 . Let $$E'$$ E ′ denote the Köthe dual of E and $$\tau (E,E')$$ τ ( E , E ′ ) stand for the natural Mackey topology on E. It is shown that every nuclear operator $$T:E\rightarrow X$$ T : E → X between the locally convex space $$(E,\tau (E,E'))$$ ( E , τ ( E , E ′ ) ) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X between the locally convex space $$(L^\infty ,\tau (L^\infty ,L^1))$$ ( L ∞ , τ ( L ∞ , L 1 ) ) and a Banach space X is nuclear if and only if its representing measure $$m_T:\Sigma \rightarrow X$$ m T : Σ → X has the Radon-Nikodym property and $$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$ | m T | ( Ω ) = ‖ T ‖ nuc (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on $$L^\infty $$ L ∞ are nuclear. Moreover, it is shown that every nuclear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X admits a factorization through some Orlicz space $$L^\varphi $$ L φ , that is, $$T=S\circ i_\infty $$ T = S ∘ i ∞ , where $$S:L^\varphi \rightarrow X$$ S : L φ → X is a Bochner representable and compact operator and $$i_\infty :L^\infty \rightarrow L^\varphi $$ i ∞ : L ∞ → L φ is the inclusion map.

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An Embedding Theorem for Separable Locally Convex Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1971-023-1 ◽

1971 ◽

Vol 14 (1) ◽

pp. 119-120 ◽

Cited By ~ 1

Author(s):

Robert H. Lohman

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Convex Space ◽

Separable Banach Space ◽

Locally Convex Space ◽

Embedding Theorem ◽

Vector Spaces ◽

Locally Convex Spaces ◽

Topological Vector Spaces ◽

Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

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Injective absolutely summing operators

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065105 ◽

1988 ◽

Vol 103 (3) ◽

pp. 497-502

Author(s):

Susumu Okada ◽

Yoshiaki Okazaki

Keyword(s):

Banach Space ◽

Uniform Convergence ◽

Convex Space ◽

Locally Convex Space ◽

Dimensional Banach Space ◽

Infinite Dimensional ◽

Infinite Dimensional Banach Space ◽

Integrable Functions ◽

Locally Convex ◽

Convergence In Mean

Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see [14]). The Archimedes integral introduced in [9] does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see [9]).

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Regularization of cylindrical processes in locally convex spaces

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025720500277 ◽

2020 ◽

pp. 2050027

Author(s):

Christian A. Fonseca-Mora

Keyword(s):

Convex Space ◽

Lévy Process ◽

Sufficient Conditions ◽

Locally Convex Space ◽

Levy Process ◽

Locally Convex Spaces ◽

Locally Convex ◽

Strong Dual ◽

Convex Spaces

Let [Formula: see text] be a locally convex space and let [Formula: see text] denote its strong dual. In this paper, we introduce sufficient conditions for the existence of a continuous or a càdlàg [Formula: see text]-valued version to a cylindrical process defined on [Formula: see text]. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we provide sufficient conditions for the existence of a [Formula: see text]-valued càdlàg Lévy process version to a given cylindrical Lévy process in [Formula: see text].

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On the holomorphical classification of spaces of holomorphic germs

Nagoya Mathematical Journal ◽

10.1017/s0027763000019565 ◽

1981 ◽

Vol 84 ◽

pp. 85-118 ◽

Cited By ~ 4

Author(s):

Jorge Aragona

Keyword(s):

Banach Space ◽

Convex Space ◽

Complex Banach Space ◽

Locally Convex Space ◽

Compact Set ◽

Locally Convex

Let K be a compact set in a complex metrizable locally convex space E and F a complex Banach space. The study of the space of holomorphic germs ℋ(K; F) endowed with the Nachbin topology was undertaken by several authors.

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Some Intersections of the Weighted -Spaces

Abstract and Applied Analysis ◽

10.1155/2013/986857 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

F. Abtahi ◽

H. G. Amini ◽

H. A. Lotfi ◽

A. Rejali

Keyword(s):

Banach Space ◽

Sufficient Conditions ◽

Locally Convex Space ◽

Locally Compact Group ◽

Weight Functions ◽

Convolution Product ◽

Arbitrary Subset ◽

Locally Compact ◽

Algebraic Properties ◽

Locally Convex

Let be a locally compact group an arbitrary family of the weight functions on and . The locally convex space as a subspace of is defined. Also, some sufficient conditions for that space to be a Banach space are provided. Furthermore, for an arbitrary subset of and a positive submultiplicative weight function on , Banach subspace of is introduced. Then some algebraic properties of , as a Banach algebra under convolution product, are investigated.

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Superdiagonal forms for related linear operators

Glasgow Mathematical Journal ◽

10.1017/s0017089500005723 ◽

1985 ◽

Vol 26 (1) ◽

pp. 19-23

Author(s):

Volker Wrobel

Keyword(s):

Banach Space ◽

Convex Space ◽

Invariant Subspaces ◽

Main Tool ◽

Complex Banach Space ◽

Locally Convex Space ◽

Linear Operators ◽

Permanence Property ◽

Locally Convex

The concept of superdiagonal forms for n × nmatrices T with complex entries has been extended by J. R. Ringrose [4] to the setting of compact linear operators T:X→X acting on a complex Banach space X. In a recent paper D. Koros [2] generalized Ringrose's approach to the case of compact linear operators T:X→X on a complex locally convex space X. The reason why both authors confine their attention to the class of compact linear operators is that the existence of proper closed invariant subspaces is, aside from Riesz-Schauder theory, the main tool in their construction. In the present paper it is shown that the existence of superdiagonal forms possesses a certain permanence property in the following sense.

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