Displaying similar documents to “Continuity of order-preserving functions”

On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

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Let Λ ⊂ ℝ n × ℝ m and k be a positive integer. Let f : ℝ n → ℝ m be a locally bounded map such that for each ( ξ , η ) ∈ Λ , the derivatives D ξ j f ( x ) : = d j d t j f ( x + t ξ ) | t = 0 , j = 1 , 2 , ⋯ k , exist and are continuous. In order to conclude that any such map f is necessarily of class C k it is necessary and sufficient that Λ be not contained in the zero-set of a nonzero homogenous polynomial Φ ( ξ , η ) which is linear in η = ( η 1 , η 2 , ⋯ , η m ) and homogeneous of degree k in ξ = ( ξ 1 , ξ 2 , ⋯ , ξ n ) . This generalizes a result of J. Boman for the case k = 1 . The statement and the proof of a theorem of Boman for the case k = ∞ is...

Generalized M-norms on ordered normed spaces

I. Tzschichholtz, M. R. Weber (2005)

Banach Center Publications

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L-norms and M-norms on Banach lattices, unit-norms and base norms on ordered vector spaces are well known. In this paper m- and m ≤ -norms are introduced on ordered normed spaces. They generalize the notions of the M-norm and the order-unit norm, possess also some interesting properties and can be characterized by means of the constants of reproducibility of cones. In particular, the dual norm of an ordered Banach space with a closed cone turns out to be additive on the dual cone if and...

Some remarks on the product of two C α -compact subsets

Salvador García-Ferreira, Manuel Sanchis, Stephen W. Watson (2000)

Czechoslovak Mathematical Journal

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For a cardinal α , we say that a subset B of a space X is C α -compact in X if for every continuous function f X → ℝ α , f [ B ] is a compact subset of ℝ α . If B is a C -compact subset of a space X , then ρ ( B , X ) denotes the degree of C α -compactness of B in X . A space X is called α -pseudocompact if X is C α -compact into itself. For each cardinal α , we give an example of an α -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

Counting linearly ordered spaces

Gerald Kuba (2014)

Colloquium Mathematicae

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For a transfinite cardinal κ and i ∈ 0,1,2 let ℒ i ( κ ) be the class of all linearly ordered spaces X of size κ such that X is totally disconnected when i = 0, the topology of X is generated by a dense linear ordering of X when i = 1, and X is compact when i = 2. Thus every space in ℒ₁(κ) ∩ ℒ₂(κ) is connected and hence ℒ₁(κ) ∩ ℒ₂(κ) = ∅ if κ < 2 ℵ ₀ , and ℒ₀(κ) ∩ ℒ₁(κ) ∩ ℒ₂(κ) = ∅ for arbitrary κ. All spaces in ℒ₁(ℵ₀) are homeomorphic, while ℒ₂(ℵ₀) contains precisely ℵ₁ spaces up to homeomorphism. The...

On the positivity of semigroups of operators

Roland Lemmert, Peter Volkmann (1998)

Commentationes Mathematicae Universitatis Carolinae

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In a Banach space E , let U ( t ) ( t > 0 ) be a C 0 -semigroup with generating operator A . For a cone K ⊆ E with non-empty interior we show: ( ☆ )     U ( t ) [ K ] ⊆ K ( t > 0 ) holds if and only if A is quasimonotone increasing with respect to K . On the other hand, if A is not continuous, then there exists a regular cone K ⊆ E such that A is quasimonotone increasing, but ( ☆ ) does not hold.