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K. F. Ng

Bio: K. F. Ng is an academic researcher. The author has contributed to research in topics: Dual norm & Banach space. The author has an hindex of 1, co-authored 1 publications receiving 3 citations.

Papers
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Journal ArticleDOI
TL;DR: In this article, the authors give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is a-monotone (a > 1) if and only if for each / in B* there exists g 6 B* with g > 0, / and ||g|| < aN(f).
Abstract: Let B be an ordered Banach space with ordered Banach dual space. Let N denote the canonical half-norm. We give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is a-monotone (a > 1) if and only if for each / in B* there exists g 6 B* with g > 0, / and ||g|| < aN(f). We also establish a dual result characterizing a-monotonicity of B'.

3 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, the authors give an overview of normality and conormality properties of pre-ordered Banach spaces and define a class of ordered spaces called quasi-lattices which strictly contain the Banach lattices, and prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi lattice.
Abstract: We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces \(X\) and \(Y\) with closed cones we investigate normality of \(B(X,Y)\) in terms of normality and conormality of the underlying spaces \(X\) and \(Y\). Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples \(X\) and \(Y\) that are not Banach lattices, but for which \(B(X,Y)\) is normal. In particular, we show that a Hilbert space \(\mathcal {H}\) endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if \(\dim \mathcal {H}\ge 3\)), and satisfies an identity analogous to the elementary Banach lattice identity \(\Vert |x|\Vert =\Vert x\Vert \) which holds for all elements \(x\) of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.

17 citations

Journal ArticleDOI
TL;DR: In this article, the authors give an overview of normality and conormality properties of pre-ordered Banach spaces and define quasi-lattices which strictly contain the Banach lattices, and prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi lattice.
Abstract: We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the underlying spaces $X$ and $Y$. Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $X$ and $Y$ that are not Banach lattices, but for which $B(X,Y)$ is normal. In particular, we show that a Hilbert space $\mathcal{H}$ endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $\dim\mathcal{H}\geq3$), and satisfies an identity analogous to the elementary Banach lattice identity $\||x|\|=\|x\|$ which holds for all elements $x$ of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.

3 citations

Book ChapterDOI
01 Aug 2019
TL;DR: For any Banach space X, ordered by a closed generating cone C ⊆ X, do there always exist Lipschitz functions ⋅+ : X → C and ⋆− : x → C satisfying x = x+ − x− for every x ∈ X?
Abstract: Consider the following still-open problem: for any Banach space X, ordered by a closed generating cone C ⊆ X, do there always exist Lipschitz functions ⋅+ : X → C and ⋅− : X → C satisfying x = x+ − x− for every x ∈ X?