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.

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).

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

scholarly journals Essential supremum norm differentiability

1985 ◽  
Vol 8 (3) ◽  
pp. 433-439
Author(s):  
I. E. Leonard ◽  
K. F. Taylor
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Real Banach Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Supremum Norm ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Fréchet Differentiability ◽  
Finite Measure Space

The points of Gateaux and Fréchet differentiability inL∞(μ,X)are obtained, where(Ω,∑,μ)is a finite measure space andXis a real Banach space. An application of these results is given to the spaceB(L1(μ,ℝ),X)of all bounded linear operators fromL1(μ,ℝ)intoX.

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.

scholarly journals Best approximation in Orlicz spaces

1991 ◽  
Vol 14 (2) ◽  
pp. 245-252 ◽  
Author(s):  
H. Al-Minawi ◽  
S. Ayesh
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Best Approximation ◽  
Measure Space ◽  
Real Banach Space ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Finite Measure ◽  
Reflexive Subspace ◽  
Finite Measure Space

LetXbe a real Banach space and(Ω,μ)be a finite measure space andϕbe a strictly icreasing convex continuous function on[0,∞)withϕ(0)=0. The spaceLϕ(μ,X)is the set of all measurable functionsfwith values inXsuch that∫Ωϕ(‖c−1f(t)‖)dμ(t)<∞for somec>0. One of the main results of this paper is: “For a closed subspaceYofX,Lϕ(μ,Y)is proximinal inLϕ(μ,X)if and only ifL1(μ,Y)is proximinal inL1(μ,X)′​′. As a result ifYis reflexive subspace ofX, thenLϕ(ϕ,Y)is proximinal inLϕ(μ,X). Other results on proximinality of subspaces ofLϕ(μ,X)are proved.

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].

Applications of Variants of the Hölder Inequality and its Inverses: Extensions of Barnes, Marshall-Olkin, and Nehari Inequalities.

1978 ◽  
Vol 21 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Chung-Lie Wang
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Hölder Inequality ◽  
Bounded Subset ◽  
Real Numbers ◽  
Integrable Functions ◽  
Finite Measure Space

The primary aim of this paper is to extend Barnes [1], Marshall-Olkin [6], and Nehari [8] inequalities as applications of some results introduced in [10] by the author.Since several results from various sources are adopted here, a unified notation is required in order to simplify our subsequent arguments. To this end, let Lp = Lp(S, ∑, μ), p>0 (unless otherwise stated), be the space of all pth power non-negative integrable functions over a given finite measure space (S, ∑, μ) (where S may be regarded as a bounded subset of real numbers).

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 Interchanging a limit and an integral: necessary and sufficient conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Takashi Kamihigashi
Keyword(s):  
Measure Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Integrable Functions ◽  
Finite Measure Space ◽  
Pointwise Limit ◽  
Necessary And Sufficient

AbstractLet $\{f_{n}\}_{n \in \mathbb {N}}$ { f n } n ∈ N be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$ ( Ω , F , μ ) . Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$ lim n ↑ ∞ f n exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$ lim n ↑ ∞ ∫ f n d μ = ∫ lim n ↑ ∞ f n d μ .

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