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Convex transformations with Banach lattice range.

Roman Ger (1987)

Stochastica

A closed epigraph theorem for Jensen-convex mappings with values in Banach lattices with a strong unit is established. This allows one to reduce the examination of continuity of vector valued transformations to the case of convex real functionals. In particular, it is shown that a weakly continuous Jensen-convex mapping is continuous. A number of corollaries follow; among them, a characterization of continuous vector-valued convex transformations is given that answers a question raised by Ih-Ching...

Convex-compact sets and Banach discs

I. Monterde, Vicente Montesinos (2009)

Czechoslovak Mathematical Journal

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual E ' of a locally convex space E is the σ ( E ' , E ) -closure of the union of countably many σ ( E ' , E ) -relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.

Convexity around the Unit of a Banach Algebra

Kadets, Vladimir, Katkova, Olga, Martín, Miguel, Vishnyakova, Anna (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 46B20. Secondary: 46H99, 47A12.We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A showing that inf {max±||1 ± x|| − 1 : x ∈ A, ||x||=ε} ≥ (π/4e)ε²+o(ε²) as ε → 0. We also give a characterization of two-dimensional subspaces of Banach algebras containing the identity in terms of polynomial inequalities.

Convex-like inequality, homogeneity, subadditivity, and a characterization of L p -norm

Janusz Matkowski, Marek Pycia (1995)

Annales Polonici Mathematici

Let a and b be fixed real numbers such that 0 < mina,b < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that l i m s u p t → 0 + f ( t ) ≤ 0 must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the L p -norm.

Convolution algebras with weighted rearrangement-invariant norm

R. Kerman, E. Sawyer (1994)

Studia Mathematica

Let X be a rearrangement-invariant space of Lebesgue-measurable functions on ℝ n , such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on ℝ n , define X ( w ) = F : ℝ n → ℂ : ∞ > ∥ F ∥ X ( w ) : = ∥ F w ∥ X . We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at x ∈ ℝ n by ( F ∗ G ) ( x ) = ʃ ℝ n F ( x - y ) G ( y ) d y ; more precisely, when ∥ F ∗ G ∥ X ( w ) ≤ ∥ F ∥ X ( w ) ∥ G ∥ X ( w ) for all F,G ∈ X(w).

Convolution equations in the space of Laplace distributions

Maria E. Pliś (1998)

Annales Polonici Mathematici

A formal solution of a nonlinear equation P(D)u = g(u) in 2 variables is constructed using the Laplace transformation and a convolution equation. We assume some conditions on the characteristic set Char P.

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