Metrizability of the strong dual: equivalent topological characterizations

Firstly, we present some characterizations of L-fuzzy posetand their corresponding proofs. Then, we give the definition of L-fuzzy upper set and prove its related properties. Next,we define the concept of L-fuzzy Co-Alexandrov topological space on an L-fuzzy poset (X,e). We further study the properties of L-fuzzy Co-Alexandrov topological space and equivalent characterizations, which are generalizations ofthe Alexandrov topological space on a poset (X,

Finitely chainable and totally bounded metric spaces: Equivalent characterizations

Let E, Ê, and E′ denote a locally convex linear Hausdorff space, completion of E and the dual of E, respectively. It has been observed that Ê is a subspace of E″ under certain conditions on E. It is the primary goal of this paper to give necessary and sufficient conditions for the Ê ⊂ E″ to be valid. Such conditions are found and are given Theorem 4. With a variation of the technique used, several equivalent characterizations of semi-reflexive spaces are given in Theorem 5. The nationa throughtout will follow that in [2].

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and also the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.

Let K be a cone of a linear space X and Y a sequentially complete locally convex linear topological Hausdorff space. Let f : K → Y and g: K→ Y satisfy $${\alpha }^{-1}f(\alpha x) - g(x) \in U,\quad \alpha \in A, x \in K,$$ where U is a bounded subset of Y and A ⊂ [1, ∞). Under some additional assumptions we prove that there exists exactly one positively homogeneous function F : K → Y such that the differences F − f and F ;− g are bounded on K, i.e. the equation of homogeneous functions is stable in the Ulam–Hyers sense.

Tensor Products of Silva-Holomorphic Functions

Let $E$ be a locally convex topological Hausdorff space, $K$ a nonempty compact convex subset of $E$, $\mu$ a regular Borel probability measure on $E$ and $\gamma >0$. We say that the measure $\mu$ $\gamma $-represents a point $x\in K$ if $\sup_{\| f\|\le

Existence Results for Nonlinear Integral Equations

The expectation monad in quantum foundations

The Approximation Property for Certain Spaces of Holomorphic Mappings

Let ƒ be a function defined on a cone S with the values in a sequentially complete locally convex linear topological Hausdorff space Y. If there exist a bounded subset V of Y and an open interval (a, b) ⊂ (1,∞) such that for all x ∈ S and every A ∈ (a, b) the condition λ−1 ƒ(λx) − ƒ(x) ∈ V holds, then there exists a unique positively homogeneous mapping F: S → Y such that the difference F(x) − ƒ(x) is uniformly bounded on S.

On integral equations of Urysohn–Volterra type

The purpose of this paper is to investigate $\alpha $-state filters in a state residuated lattice. First, the notion of $\alpha $-state filters in a state residuated lattice is introduced. It is proved that the set of all $\alpha $-state filters in a state residuated lattice forms a complete Heyting algebra. Furthermore, the concept of quasicomplemented state residuated lattices is presented. Some equivalent conditions are derived for quasicomplemented state residuated lattices. Finally, the hull-kernel topology on the set of all prime $\alpha $-state filters in a state residuated lattice is investigated. It is demonstrated that the set of all prime $\alpha $-state filters under the hull-kernel topology is a spectral space and some certain conditions are given for the space to be either a $T_{1}$-space or a Hausdorff space. In addition, some topological characterizations are summarized for a state residuated lattice to be quasicomplemented.

Spaces of Continuous Functions

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a locally compact Hausdorff space and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 0 left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_0(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Banach space of all complex valued continuous functions vanishing at infinity in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , provided with the supremum norm. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a locally convex Hausdorff space (briefly, an lcHs) which is quasicomplete. A simple proof of the Grothendieck theorem on the Dieudonné property of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 0 left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_0(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given. The present proof is much simpler than that given in an earlier work of the author ( <italic> Characterizations of weakly compact operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 0 left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_0(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> </italic> , Trans. Amer. Math. Soc. <bold>350</bold> (1998), 4849-4867).