Associate space with respect to a locally $$\sigma $$ σ -finite measure on a $$\delta $$ δ -ring and applications to spaces of integrable functions defined by a vector measure

Positivity ◽  
2015 ◽  
Vol 20 (3) ◽  
pp. 515-539 ◽  
Author(s):  
Celia Avalos-Ramos ◽  
Fernando Galaz-Fontes
Keyword(s):  
Vector Measure ◽  
Finite Measure ◽  
Integrable Functions ◽  
Associate Space

scholarly journals Spaces of integrable functions with respect to a vector measure and factorizations through Lp and Hilbert spaces

2007 ◽  
Vol 330 (2) ◽  
pp. 1249-1263 ◽  
Author(s):  
A. Fernández ◽  
F. Mayoral ◽  
F. Naranjo ◽  
C. Sáez ◽  
E.A. Sánchez-Pérez
Keyword(s):  
Hilbert Spaces ◽  
Vector Measure ◽  
Integrable Functions

Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series

2007 ◽  
Vol 257 (2) ◽  
pp. 381-402 ◽  
Author(s):  
J. M. Calabuig ◽  
F. Galaz-Fontes ◽  
E. Jiménez-Fernández ◽  
E. A. Sánchez Pérez
Keyword(s):  
Vector Measure ◽  
Unconditional Convergence ◽  
Convergence Of Series ◽  
Integrable Functions

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.

scholarly journals On sequences of integrable functions

1962 ◽  
Vol 2 (3) ◽  
pp. 295-300
Author(s):  
Basil C. Rennie
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Finite Measure ◽  
Complex Banach Space ◽  
The Real ◽  
Integrable Functions ◽  
Necessary And Sufficient ◽  
Sequence Of Functions

Let f1(x), f2(x), … be a sequence of functions belonging to the real or complex Banach space L, (see S. Banach: [1] (The results can be generalised to functions on any space that is the union of countably many sets of finite measure). We are concerned with various properties that such a sequence may have, and in particular with the more important kinds of convergence (strong, weak and pointwise). This article shows what relations connect the various properties considered; for instance that for strong convergence (i.e. ║fn — f║ → 0) it is necessary and sufficient firstly that the sequence should converge weakly (i.e. if g is bounded and measurable then f(fn(x) — f(x))g(x)dx → 0) and secondly that any sub-sequence should contain a sub-sub-sequence converging p.p. to f(x).

scholarly journals Multiplication operators on spaces of integrable functions with respect to a vector measure

2008 ◽  
Vol 343 (1) ◽  
pp. 514-524 ◽  
Author(s):  
R. del Campo ◽  
A. Fernández ◽  
I. Ferrando ◽  
F. Mayoral ◽  
F. Naranjo
Keyword(s):  
Vector Measure ◽  
Multiplication Operators ◽  
Integrable Functions

Commutative Sequences of Integrable Functions and Best Approximation With Respect to the Weighted Vector Measure Distance

2005 ◽  
Vol 54 (4) ◽  
pp. 495-510 ◽  
Author(s):  
L. M. García. Raffi ◽  
E. A. Sánchez. Pérez ◽  
J. V. Sánchez. Pérez
Keyword(s):  
Best Approximation ◽  
Vector Measure ◽  
Integrable Functions

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 Remarks on Weak Compactness in L1(μ,X)

1977 ◽  
Vol 18 (1) ◽  
pp. 87-91 ◽  
Author(s):  
J. Diestel
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Factorization Method ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Integrable Functions ◽  
Nikodym Property ◽  
Potential Applicability ◽  
Image Position

Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L1 (μ,X) the Banach space of (equivalence classes of) μ-strongly measurable X-valued Bochner integrable functions f:Ω→X normed byThe problem of characterizing the relatively weakly compact subsets of L1(Ω, X) remains open. It is known that for a bounded subset of L1(μ, X) to be relatively weakly compact it is necessary that the set be uniformly integrable; recall that K ⊆ L1, (μ, X) is uniformly integrable whenever given ε >0 there exists δ > 0 such that if μ (E) ≦ δ then ∫E∥f∥ dμ ≦ δ, for all f ∈ K. S. Chatterji has noted that in case X is reflexive this condition is also sufficient [4]. At present unless one assumes that both X and X* have the Radon-Nikodym Property (see [1]), a rather severe restriction which, for purposes of potential applicability, is tantamount to assuming reflexivity, no good sufficient conditions for weak compactness in L1(μ, X) exist. This note puts forth such sufficient conditions; the basic tool is the recent factorization method of W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski [3].

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