A coercive James’s weak compactness theorem and nonlinear variational problems

Orihuela, J.; Galán, M. Ruiz

doi:10.1016/j.na.2011.08.062

Cited by 16 publications

(10 citation statements)

References 12 publications

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“…The separation theorem provides x * * 0 ∈ B E * * and α < β satisfying (17) x * * 0 , x * < α < β < x * * 0 , x * 0 for every x * ∈ co(B) + Λ D · .…”

Section: Unbounded Nonlinear Weak * -James' Theoremmentioning

confidence: 99%

“…Our results here are nonlinear analogues to those of [2,9] for weak * -James compactness results. This variational setting began with [17,18,22] and it was used to deal with robust representation of risk measures in mathematical finance. A main contribution reads as follows: Then the map f is weak * -lower semicontinuous and for every µ ∈ R, the sublevel set f −1 ((−∞, µ]) is weak * -compact.…”

Section: Unbounded Nonlinear Weak * -James' Theoremmentioning

confidence: 99%

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One-sided James' compactness theorem

Cascales

Orihuela

Pérez

2017

Journal of Mathematical Analysis and Applications

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We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let E be a Banach space such that (B E * , ω * ) is convex block compact. Let A and B be bounded, closed and convex sets withattains its infimum on A and its supremum on B, then A and B are both weakly compact.We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.2010 Mathematics Subject Classification. 46A50, 46B50.

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“…The separation theorem provides x * * 0 ∈ B E * * and α < β satisfying (17) x * * 0 , x * < α < β < x * * 0 , x * 0 for every x * ∈ co(B) + Λ D · .…”

Section: Unbounded Nonlinear Weak * -James' Theoremmentioning

confidence: 99%

Section: Unbounded Nonlinear Weak * -James' Theoremmentioning

confidence: 99%

One-sided James' compactness theorem

Cascales

Orihuela

Pérez

2017

Journal of Mathematical Analysis and Applications

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“…Since ϕ is finite-valued, this shows that lim Z 1 →∞ ϕ * (Z)/ Z 1 = ∞ (i.e., ϕ * is coercive). Then the implication (3) ⇒ (2) follows from coercive James's theorem due to [26] (recalled below as Theorem 5.2).…”

Section: Monotone Convex Functions On L ∞mentioning

confidence: 99%

“…ϕ(X + c) = ϕ(X) + c if c ∈ R), it was first obtained by [21] with an additional separability assumption, and the latter assumption was removed later by [10] using a homogenization trick. See also [26,27].…”

Section: Monotone Convex Functions On L ∞mentioning

confidence: 99%

Maximum Lebesgue extension of monotone convex functions

Owari

2014

Journal of Functional Analysis

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Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a "nice" dual representation of the function. arXiv:1304.7934v2 [math.FA] 19 Jan 2014 2 K. OwariAs a first (trivial) example, we briefly see what happens when ϕ 0 is linear.Example 1.2. Let ϕ 0 be a positive (monotone) linear functional on L ∞ . Then it is finite-valued and identified with a finitely additive measure ν 0 (A) := ϕ 0 (1 A ) as ϕ 0 (X) = Ω Xdν 0 , while (1.2) is equivalent to saying that ν 0 is σ-additive. If the latter is the case, the "usual" integralφ(X) := Ω Xdν 0 defines a Lebesgue-preserving extension of ϕ 0 to L 1 (ν 0 ) := {X ∈ L 0 : Ω |X|dν 0 < ∞}. On the other hand, if ϕ is a monotone convex function on a solid space X ⊂ L 0 with (1.1) and ϕ| L ∞ = ϕ 0 , it is easy that ϕ must be positive, linear and finite on X . Then |X|dν 0 = lim nφ (|X| ∧ n) = lim n ϕ 0 (|X| ∧ n) = lim n ϕ(|X| ∧ n) = ϕ(|X|) < ∞ if X ∈ X , hence X ⊂ L 1 (ν 0 ), where the first equality follows from the monotone convergence theorem, and the fourth from the Lebesgue property of ϕ on X . Similarly, but with X1 {|X|≤n} instead of |X| ∧ n, we see also that ϕ =φ| X . Namely, (φ, L 1 (ν 0 )) is the maximum Lebesgue-preserving extension of ϕ 0 . ♦

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“…The next preparatory result is a version of James's theorem obtained by [14] which we shall use in the proof of (3) ⇒ (2). Theorem 2.2 ([14], Theorem 2).…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

On the Lebesgue property of monotone convex functions

Owari

2013

Math Finan Econ

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The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous linear functionals. This generalizes and unifies several recent results obtained in the context of convex risk measures.

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A coercive James’s weak compactness theorem and nonlinear variational problems

Cited by 16 publications

References 12 publications

One-sided James' compactness theorem

One-sided James' compactness theorem

Maximum Lebesgue extension of monotone convex functions

On the Lebesgue property of monotone convex functions

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