scholarly journals Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Marian Nowak
Keyword(s):  
Banach Space ◽  
Vector Lattice ◽  
Topological Properties ◽  
Complex Vector ◽  
Mackey Topology ◽  
Measurable Functions ◽  
Complex Valued ◽  
Locally Convex

Let Σ be aσ-algebra of subsets of a nonempty set Ω. Let be the complex vector lattice of bounded Σ-measurable complex-valued functions on Ω and let be the Banach space of all bounded countably additive complex-valued measures on Ω. We study locally solid topologies on . In particular, it is shown that the Mackey topology is the finest locally convex-solidσ-Lebesgue topology on .

Absolute Convergence Factors for Hp Series

1969 ◽  
Vol 21 ◽  
pp. 187-195 ◽  
Author(s):  
James Caveny
Keyword(s):  
Banach Space ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Famous Theorem ◽  
Conjugate Space ◽  
Measurable Functions ◽  
Image Position ◽  
Complex Valued

A famous theorem of Hardy asserts that if f ∊ H1, then the sequence of Fourier coefficients satisfies . For this reason we say that the sequence (1, 1/2, 1/3, …) belongs to the multiplier class (H1, l1). In this paper, we investigate the multiplier classes (Hp, l1) for 1 ≧ p ≧ ∞. Our observations are based on the fact that a sequence (λ(0), λ(l), …) belongs to (Hp, l1) independent of the arguments of its terms. We also show that (Hp, l1) may be thought of as the conjugate space of a certain Banach space.1. Preliminaries.Lp denotes the space of complex-valued Lebesgue measurable functions f defined on the circle |z| = 1 such thatis finite.

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 Connectedness in some topological vector-lattice groups of sequences

2010 ◽  
Vol 107 (1) ◽  
pp. 150 ◽  
Author(s):  
Lech Drewnowski ◽  
Marek Nawrocki
Keyword(s):  
Banach Space ◽  
Vector Lattice ◽  
Connected Component ◽  
Similar Nature

Let $\eta$ be a strictly positive submeasure on $\mathsf N$. It is shown that the space $\omega(\eta)$ of all real sequences, considered with the topology $\tau_{\eta}$ of convergence in submeasure $\eta$, is (pathwise) connected iff $\eta$ is core-nonatomic. Moreover, for an arbitrary submeasure $\eta$, the connected component of the origin in $\omega(\eta)$ is characterized and shown to be an ideal. Some results of similar nature are also established for general topological vector-lattice groups as well as for the topological vector groups of Banach space valued sequences with the topology $\tau_{\eta}$.

scholarly journals POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

scholarly journals Nonsmooth analysis and optimization on partially ordered vector spaces

1992 ◽  
Vol 15 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Thomas W. Reiland
Keyword(s):  
Vector Lattice ◽  
Vector Spaces ◽  
Range Space ◽  
Topological Vector Spaces ◽  
Complete Vector Lattice ◽  
Lipschitz Mappings ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Complete Vector ◽  
Locally Convex

Interval-Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz-type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.

The Isometries of Hp

1964 ◽  
Vol 16 ◽  
pp. 721-728 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Circle ◽  
Lebesgue Measure ◽  
Measurable Functions ◽  
Image Position ◽  
Complex Valued

Let a be the Lebesgue measure on the unit circle |z| = 1 withand let Lp be the space of complex-valued σ-measurable functions f such thatis finite. Hp is the closure in Lp of the algebra of analytic polynomials

scholarly journals Best Simultaneous approximation of quasi-continuous functions by monotone functions

1991 ◽  
Vol 50 (3) ◽  
pp. 391-408 ◽  
Author(s):  
Salem M. A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Continuous Functions ◽  
Unit Interval ◽  
Closed Convex Cone ◽  
Monotone Functions ◽  
Measurable Functions ◽  
Best Simultaneous Approximation ◽  
Nondecreasing Functions

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

The β- and γ-duality of sequence spaces

1967 ◽  
Vol 63 (4) ◽  
pp. 963-981 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Linear Space ◽  
Sequence Spaces ◽  
Topological Properties ◽  
Product Topology ◽  
Convex Topology ◽  
Locally Convex Topology ◽  
Locally Convex

A K-space (E, τ) is a linear space E of sequences with a locally convex topology τ for which the inclusion map: (E, τ) → (ω, product topology) is continuous. In (2) topological properties of K-spaces were determined directly from properties of the space E and the topology τ. It is, however, very natural to consider duality properties of K-spaces and the purpose of this paper is to determine some of these properties.

scholarly journals Epicompletetion of archimedean l–groups and vector lattices with weak unit

1990 ◽  
Vol 48 (1) ◽  
pp. 25-56 ◽  
Author(s):  
Richard N. Ball ◽  
Anthony W. Hager
Keyword(s):  
Vector Lattice ◽  
Boolean Algebras ◽  
Weak Order ◽  
Minimal Extension ◽  
Order Unit ◽  
Weak Unit ◽  
Vector Lattices ◽  
Measurable Functions ◽  
Baire Functions

AbstractIn the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

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