scholarly journals The Mazur property and completeness in the space of Bochner-integrable functions L1(μ, X)

1992 ◽  
Vol 34 (2) ◽  
pp. 201-206
Author(s):  
G. Schlüchtermann
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Positive Measure ◽  
Dual Space ◽  
Measure Theory ◽  
Locally Convex Space ◽  
Continuous Functional ◽  
Injective Tensor Product ◽  
Integrable Functions ◽  
Locally Convex

A locally convex space (E, ) has the Mazur Property if and only if every linear -sequential continuous functional is -continuous (see [11]).In the Banach space setting, a Banach space X is a Mazur space if and only if the dual space X* endowed with the w*-topology has the Mazur property. The Mazur property was introduced by S. Mazur, and, for Banach spaces, it is investigated in detail in [4], where relations with other properties and applications to measure theory are listed. T. Kappeler obtained (see [8]) certain results for the injective tensor product and showed that L1(μ, X), the space of Bochner-integrable functions over a finite and positive measure space (S, σ, μ), is a Mazur space provided X is also, and ℓ1 does not embed in X.

scholarly journals Injective absolutely summing operators

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]).

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
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.

scholarly journals Nuclear operators on Banach function spaces

Positivity ◽  
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.

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
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.

scholarly journals Best simultaneous approximation on metric spaces via monotonous norms

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3777-3787
Author(s):  
Mona Khandaqji ◽  
Aliaa Burqan
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Finite Number ◽  
Metric Space ◽  
Measure Space ◽  
Simultaneous Approximation ◽  
Metric Spaces ◽  
Closed Subspace ◽  
Integrable Functions ◽  
Best Simultaneous Approximation

For a Banach space X, L?(T,X) denotes the metric space of all X-valued ?-integrable functions f : T ? X, where the measure space (T,?,?) is a complete positive ?-finite and ? is an increasing subadditive continuous function on [0,?) with ?(0) = 0. In this paper we discuss the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y?X to a finite number of elements in X.

Necessary and Sufficient Conditions for the Equality of L(f) and l1

1982 ◽  
Vol 34 (2) ◽  
pp. 406-410 ◽  
Author(s):  
Waleed Deeb
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Locally Convex Space ◽  
Natural Question ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Infinite Dimensional Subspace

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.

scholarly journals Best N-Simultaneous Approximation in Lp(μ,X)

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Tijani Pakhrou
Keyword(s):  
Banach Space ◽  
Compact Subset ◽  
Measure Space ◽  
Simultaneous Approximation ◽  
Finite Measure ◽  
Weakly Compact ◽  
Integrable Functions ◽  
Finite Measure Space

Let X be a Banach space. Let 1≤p<∞ and denote by Lp(μ,X) the Banach space of all X-valued Bochner p-integrable functions on a certain positive complete σ-finite measure space (Ω,Σ,μ), endowed with the usual p-norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset W of X, the set Lp(μ,W) is N-simultaneously proximinal in Lp(μ,X) for any arbitrary monotonous norm N in Rn.

Complemented copies of l1, in Lp(μ;E)

1992 ◽  
Vol 111 (3) ◽  
pp. 531-534 ◽  
Author(s):  
José Mendoza
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Finite Measure ◽  
Complemented Subspace ◽  
Integrable Functions ◽  
Usual Norm ◽  
Finite Measure Space

AbstractLet E be a Banach space, let (Ω, Σ, μ) a finite measure space, let 1 < p < ∞ and let Lp(μ;E) the Banach space of all E-valued p-Bochner μ-integrable functions with its usual norm. In this note it is shown that E contains a complemented subspace isomorphic to l1 if (and only if) Lp(μ; E) does. An extension of this result is also given.

scholarly journals Weak star separability

1979 ◽  
Vol 20 (2) ◽  
pp. 253-257 ◽  
Author(s):  
E.N. Dancer ◽  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Locally Convex Spaces ◽  
Weak Star ◽  
Locally Convex ◽  
Convex Spaces

For a Banach space X, Susumu Okada raised the question of whether the unit hall of the dual space X* is weak* separable if X* is weak* separable. The problem occurred in the theory of manifolds modelled on locally convex spaces. We answer the question in the negative but show that it is true for particular types of spaces.

Contractions with Fixed Points and Conditional Expectation

1975 ◽  
Vol 18 (4) ◽  
pp. 475-478
Author(s):  
A. N. Al-Hussaini
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Simple Form ◽  
Conditional Expectation ◽  
Measure Space ◽  
Finite Measure ◽  
Integrable Functions ◽  
Finite Measure Space ◽  
Bounded Functions ◽  
The Given

Let (Ω, α, μ) be a σ-finite measure space. By Lp(Ω, α, μ) or Lp for short we denote the usual Banach space of pth power μ-integrable functions on Ω if 1≤p<+ ∞ and μ-essentially bounded functions on Ω, if p= +∞. In section (2) we characterize conditional expectation, by a method different than those used previously. Modulus of a given contraction is discussed in section (3). If the given contraction has a fixed point, then its modulus has a simple form (theorem 3.2).

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