Extension of linear functionals in Banach spaces of measurable functions

1976 ◽  
Vol 20 (5) ◽  
pp. 969-973
Author(s):  
M. S. Braverman ◽  
G. Ya. Lozanovskii
Keyword(s):  
Banach Spaces ◽  
Linear Functionals ◽  
Measurable Functions

scholarly journals Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Qp over Primes p

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 199 ◽  
Author(s):  
Ilwoo Cho ◽  
Palle Jorgensen
Keyword(s):  
Probability Space ◽  
Free Probability ◽  
Number Fields ◽  
Linear Functionals ◽  
Real Numbers ◽  
Probabilistic Information ◽  
Closed Intervals ◽  
Measurable Functions ◽  
Semicircular Elements

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space ( LS , τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p . In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated” linear functionals on LS .

scholarly journals On geometric properties of ⊥BJCϵ symmetric in some types of Banach spaces

2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Geometric Properties ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.

The Hahn-Banach theorem for non-Archimedean-valued fields

1952 ◽  
Vol 48 (1) ◽  
pp. 41-45 ◽  
Author(s):  
A. W. Ingleton
Keyword(s):  
Banach Spaces ◽  
Normed Linear Space ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Necessary Condition ◽  
Linear Functionals ◽  
Banach Theorem ◽  
Necessary And Sufficient ◽  
Special Case

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.

The Convergence of Series for Various Choices of Sign in Banach Spaces

1975 ◽  
Vol 27 (2) ◽  
pp. 475-480 ◽  
Author(s):  
James Shirey
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Series Expansion ◽  
Norm Convergence ◽  
Convergence Of Series ◽  
Measurable Functions ◽  
Haar Series

1. Let (xn, Xn) denote a basis for a Banach space (X, ∥ • ∥) of measurable functions in (0, 1).It is shown in [2] and [9] that the equivalence of the normsand ∥ • ∥ is equivalent to the unconditionality of the basis (xn, Xn). In [8] a weaker relationship between these norms is exploited to establish the existence of an element of L1(E) for each E ⊂ (0, 1), |£| > 0, whose Haar series expansion is conditionally convergent in the norm of L\(E).In this note, a Lemma of Orlicz [7] is generalized to provide a relationship between , and the changes in sign that are tolerated in without disruption of norm convergence.

scholarly journals Quasi reflexivity and the sup of linear functionals

1997 ◽  
Vol 55 (1) ◽  
pp. 89-98
Author(s):  
P.K. Jain ◽  
K.K. Arora ◽  
D.P. Sinha
Keyword(s):  
Banach Spaces ◽  
Unit Sphere ◽  
Linear Functionals ◽  
Reflexive Banach Spaces ◽  
Bounded Linear ◽  
Asplund Spaces ◽  
Linearly Independent

Quasi reflexive Banach spaces are characterised among the weakly countably determined Asplund spaces, in terms of the cardinality of the sets of linearly independent bounded linear functionals each of which does not attain its supremum on the unit sphere.

scholarly journals Proper 1-ball contractive retractions in Banach spaces of measurable functions

2005 ◽  
Vol 72 (2) ◽  
pp. 299-315 ◽  
Author(s):  
D. Caponetti ◽  
A. Trombetta ◽  
G. Trombetta
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Measurable Functions

In this paper we consider the Wośko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k ≤ 1 for which there exists a k-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct.

scholarly journals A Gelfand-Phillips space not containing l1 whose dual ball is not weak * sequentially compact

2001 ◽  
Vol 43 (1) ◽  
pp. 125-128 ◽  
Author(s):  
Bengt Josefson
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Isomorphic Copy ◽  
Linear Functionals ◽  
Tree Space ◽  
Convergent Sequences

A set D in a Banach space E is called limited if pointwise convergent sequences of linear functionals converge uniformly on D and E is called a GP-space (after Gelfand and Phillips) if every limited set in E is relatively compact. Banach spaces with weak * sequentially compact dual balls (W*SCDB for short) are GP-spaces and l1 is a GP-space without W*SCDB. Disproving a conjecture of Rosenthal and inspired by James tree space, Hagler and Odell constructed a class of Banach spaces ([HO]-spaces) without both W*SCDB and subspaces isomorphic to l1. Schlumprecht has shown that there is a subclass of the [HO]-spaces which are also GP-spaces. It is not clear however if any [HO]-construction yields a GP-space—in fact it is not even clear that W*SCDB[lrarr ]GP-space is false in general for the class of Banach spaces containing no subspace isomorphic to l1. In this note the example of Hagler and Odell is modified to yield a GP-space without W*SCDB and without an isomorphic copy of l1.

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