Search results for "parable"
showing 10 items of 112 documents
Description of the limit set of Henstock–Kurzweil integral sums of vector-valued functions
2015
Abstract Let f be a function defined on [ 0 , 1 ] and taking values in a Banach space X . We show that the limit set I HK ( f ) of Henstock–Kurzweil integral sums is non-empty and convex when the function f has an integrable majorant and X is separable. In the same setting we give a complete description of the limit set.
A decomposition theorem for compact-valued Henstock integral
2006
We prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of Γ.
Rademacher Theorem for Fréchet spaces
2010
Abstract Let X be a separable Frechet space. In this paper we define a class A of null sets in X that is properly contained in the class of Aronszajn null sets, and we prove that a Lipschitz map from an open subset of X into a Gelfand-Frechet space is Gateaux differentiable outside a set belonging to A. This is an extension to Frechet spaces of a result (see [PZ]) due to D. Preiss and L. Zajicek.
On the equivalence of McShane and Pettis integrability in non-separable Banach spaces
2009
Abstract We show that McShane and Pettis integrability coincide for functions f : [ 0 , 1 ] → L 1 ( μ ) , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelof determined Banach space X, a scalarly null (hence Pettis integrable) function h : [ 0 , 1 ] → X and an absolutely summing operator u from X to another Banach space Y such that the composition u ○ h : [ 0 , 1 ] → Y is not Bochner integrable; in particular, h is not McShane integrable.
Polynomial method to study the entanglement of pure N-qubit states
2009
We present a mapping which associates pure N-qubit states with a polynomial. The roots of the polynomial characterize the state completely. Using the properties of the polynomial we construct a way to determine the separability and the number of unentangled qubits of pure N-qubit states.
Scalable Ellipsoidal Classification for Bipartite Quantum States
2008
The Separability Problem is approached from the perspective of Ellipsoidal Classification. A Density Operator of dimension N can be represented as a vector in a real vector space of dimension $N^{2}- 1$, whose components are the projections of the matrix onto some selected basis. We suggest a method to test separability, based on successive optimization programs. First, we find the Minimum Volume Covering Ellipsoid that encloses a particular set of properly vectorized bipartite separable states, and then we compute the Euclidean distance of an arbitrary vectorized bipartite Density Operator to this ellipsoid. If the vectorized Density Operator falls inside the ellipsoid, it is regarded as s…
The Separable Complementation Property and Mrówka Compacta
2017
We study the separable complementation property for $C(K_{\cal A})$ spaces when $K_{\cal A}$ is the Mr\'owka compact associated to an almost disjoint family ${\cal A}$ of countable sets. In particular we prove that, if ${\cal A}$ is a generalized ladder system, then $C(K_{\cal A})$ has the separable complementation property ($SCP$ for short) if and only if it has the controlled version of this property. We also show that, when ${\cal A}$ is a maximal generalized ladder system, the space $C(K_{\cal A})$ does not enjoy the $SCP$.
ON λ-STRICT IDEALS IN BANACH SPACES
2010
AbstractWe define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.
A Decomposition of Henstock-Kurzweil-Pettis Integrable Multifunctions
2009
We proved in our earlier paper [9] that in case of separable Banach space-valued multifunctions each Henstock-Kurzweil-Pettis integrable multifunction can be represented as a sum of one of its Henstock-Kurzweil-Pettis integrable selectors and a Pettis integrable multifunction. Now, we prove that the same result can be achieved in case of an arbitrary Banach space. Applying the representation theorem we describe the multipliers of the Henstock-Kurzweil-Pettis integrable multifunctions. Then we use this description to obtain a characterization of the Henstock-Kurzweil-Pettis integrability in terms of subadditive operators.
Quasi-Normable Preduals of Spaces of Holomorphic Functions
1997
Abstract Let H ( U ) denote the space of all holomorphic functions on an open subset U of a separable Frechet space E . Let τ ω denote the compact-ported topology on H ( U ) introduced by Nachbin. Let G ( U ) denote the predual of H ( U ) constructed by Mazet. In our main result we show that E is quasi-normable if and only if G ( U ) is quasi-normable if and only if ( H ( U ), τ ω ) satisfies the strict Mackey convergence condition.