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We provide a method for witnessing nonseparability of quantum processes on shared systems, which uses the channel-state duality. The method uses a maximally entangled state as a resource. We show that using the resource provides significant advantage over the corresponding protocol for nonseparability detection without the resource., v2: 5 pages, no figures. Further gates considered as examples

The aim of this paper is to state a sharp version of the Konig supremum theorem, an equivalent reformulation of the Hahn–Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush–Kuhn–Tucker and Fritz John types, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.

The Hahn–Banach Theorem plays a crucial role in the second fundamental theorem of welfare economics. To date, all mathematical economics and advanced general equilibrium textbooks concentrate on using non-constructive or incomputable versions of this celebrated theorem. In this paper we argue for the introduction of constructive or computable Hahn–Banach theorems in mathematical economics and advanced general equilibrium theory. The suggested modification would make applied and policy-oriented economics intrinsically computational.

The conventional Hahn–Banach extension theorem based on vector space has been widely used to obtain many important and interesting results in nonlinear analysis, vector optimization and mathematical economics. Although the interval space is not a real vector space, the Hahn–Banach extension theorems based on interval spaces and nonstandard normed interval spaces can still be derived in this paper, which also shows the possible applications by considering the interval-valued problems in nonlinear analysis, vector optimization and mathematical economics.

In this paper, the existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (2) (1998) 187–200] and Theorem 4.1 in [J.W. Peng, H.W.J. Lee, W.D. Rong, X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005) 281–297], respectively, are proved by virtue of a Hahn–Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.

The classical Hahn-Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari, Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach-Alaoglu Theorem we can show that computing a Hahn-Banach extension cannot be harder than finding a zero on a compact metric space. This allows us to conclude that the Hahn-Banach extension operator is ∑20–computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn-Banach Theorem for those functionals and subspaces which admit unique extensions.

In this paper, the notion of affinelike set-valued maps is introduced and some properties of these maps are presented. Then a new Hahn–Banach extension theorem with a K -convex set-valued map dominated by an affinelike set-valued map is obtained.

The aim of this paper is to present a geometric characterization of even convexity in separable Banach spaces, which is not expressed in terms of dual functionals or separation theorems. As an application, an analytic equivalent definition for the class of evenly quasiconvex functions is derived.

In this paper, we establish some calculus rules for the limiting Frechet ϵ-subdifferentials of marginal functions and composite functions. Necessary conditions for approximate solutions of a constrained optimization problem are derived.

We establish general duality results for complete semimodules over idempotent semirings, like the max-plus semiring.

Like the extended non-negative reals R̄+ equipped with the Scott topology, there are other real topological cones such that the specialisation order yields a directed complete partially ordered set (dcpo). We will call them d-cones. Further examples are the extended probabilistic powerdomain, the set of all lower semicontinuous functions f:X→R̄+ for any topological space X and arbitrary products of given d-cones. The dual cone C∗ for a given d-cone C consists of all linear continuous functions Λ:C→R̄+. With respect to the pointwise order, addition and scalar multiplication the dual cone becomes also a d-cone. We are interested in obtaining results with our concept of d-cones that are comparable to Hahn–Banach-type theorems in functional analysis. Indeed, we can prove an Extension Theorem and a Separation Theorem for the continuous d-cones. In particular, the second implies that the elements of the dual cone C∗ separate the points of C. As a consequence of the Extension Theorem, we obtain a Sum Theorem for continuous d-cones. We will give some sufficient conditions when the previous examples of d-cones are continuous and have an additive way-below relation.

The purpose of this paper is to extend results of J. Wermer (1952, Proc. Amer. Math. Soc. 3 , 270–277), L. Brown et al. (1960, Trans. Amer. Math. Soc. 96 , 162–183), and D. Sarason (1972, J. Reine Agnew. Math. 252 , 1–15; 1966, Pacific J. Math. 17 , 511–517) on spectral subspaces of diagonalizable operators on separable complex Hilbert space to the class of so-called Jordan operators or infinite direct sums of Jordan cells.

Assuming that Ω is a nonempty set and X is a Banach space, we study the presence of a copy of c 0 in X in relation to the existence of a complemented copy of c 0 in certain closed subspaces of the Banach space l ∞ (Ω, X ) of all bounded X -valued functions defined on Ω, equipped with the supremum norm. As an application we get a result of Diaz concerning complemented copies of c 0 in L ∞ (μ, X ). A criterion for certain Banach spaces X to contain a complemented copy of c 0 whenever X contains a copy of c 0 is also included.

The main goal of this paper is to investigate the relation between fuzzy seminorms and crisp seminorms on a linear space X, and to fuzzify the analytic form of the Hahn–Banach Theorem which is one of the most important theorems of functional analysis.

Let E be a separable uniformly smooth Banach space and let A : D ( A ) ⊆ E → E be a K -positive definite operator. Let f ∈ E be arbitrary. A new iterative method is constructed which converges strongly to the unique solution of the equation Ax = f . Our work extends some of the known results due to Chidume and Osilike, and Chidume and Aneke.

Without the Hahn-Banach theorem, functional analysis would be very different from the structure we know today. Among other things, it has proved to be a very appropriate form of the Axiom of Choice for the analyst. (It is not equivalent to the Axiom of Choice, incidentally; it follows from the ultrafilter theorem which is strictly weaker.) Riesz and Helly obtained forerunners of the theorem in the turbulent mathematical world of the early 1900s. Hahn and Banach independently proved the theorem for the real case in the 1920s. Then there was Murray's extension to the complex case—easy, once you realize that ( χ ) = Re ( χ ) − iRe (i χ ). Can continuous linear maps 06 be extended as easily as linear functionals? Banach and Mazur had already proved that they could not in 1933 but it was not until Nachbin's 1950 result that a definitive answer was achieved to this more general question. In this article, we discuss the mathematical world into which the theorem entered, examine its connection to the axiom of choice, look at some ancestors, mention some of its consequences and consider some of its principal variations.

Several basic results of convexity theory are generalized to the “quantized” matrix convex sets of Wittstock. These include the Bipolar theorem, a gauge version of the Hahn–Banach theorem, and the existence theorem for support functionals.

We treat the problem of simultaneously approximating a several-times differentiable function in several variables and its derivatives by a superposition of a function, say g, in one variable. In our theory, the domain of approximation can be either compact subsets or the whole Euclidean space R d. We prove that if the domain is compact, the function g can be used without scaling, and that even in the case where the domain of approximation is the whole space R d, g can be used without scaling if it satisfies a certain condition. Moreover, g can be chosen from a wide class of functions. The basic tool is the inverse Radon transform. As a neural network can output a superposition of g, our results extend well-known neural approximation theorems which are useful in neural computation theory.

Resume Let |·| be a norm on Rn, and s a compact convex semigroup of linear |·|-contractions. Given two k-tuples of n-vectors, (x(1),…,x(k)) and (y(1),…,y(k)), we seek conditions for the existence of a contraction S ϵ s that simultaneously takes x(i) to y(i), that is, Sx(i) = y(i), for all i = 1,…,k!. Straightforward application of the separation theorem for convex sets provides a general but abstract result, in the form of a system of inequalities. Specializing |·| to the 1-norm and the ∞-norm, respectively, and s to comprise either all contractions or those contractions that preserve a particular linear form, it is possible to evaluate the characteristic functionals arising from the separation theoem. Therby the abstract results can be rduced to a tractable form, which turns out to be of the Hahn-Banach type.

The Hahn-Banach theorem is one of many fundamental results in functional analysis whose usefulness is diminished by their proof depending upon an application of the Axiom of Choice. The objection raised in this regard is not philosophical, but purely practical. When the existence of a functional has been established by this non-constructive means, the arbitrariness introduced restricts the information which can be extracted, impeding the discussion of points such as whether the functional can be shown to exist continuously with respect to a parameter [ 1, 15, 201, or equivariantly with respect to a group action [2, 333. This paper is concerned with outlining, starting from ideas which are already established, a context and a technique which allow the effects of this dependence on the Axiom of Choice to be avoided, reformulating the Hahn-Banach theorem in a way which may be applied equally to questions of continuity and of equivariance, to yield the information which is traditionally intended when applying the Hahn-Banach theorem. Before introducing the constructive context in which these ideas will be developed, consider the situations already mentioned, in which the arbitrariness of the existence of a functional established by the Axiom of Choice presents difficulties which need to be resolved. Firstly, take that of continuity in a parameter, which may be stated more precisely in the following way: consider a bundle B of seminormed spaces on a topological space X, together with a continuous mapping

Let Γ X (ʋ) = 〈 X, A (X), υ〉 be a cooperative von Neumann game with side payments, where X is a nonempty set of arbitrary cardinality, A(X) the Boolean ring generated from P(X) with the operations Δ and ∩ for addition and multiplication, respectively, such that S2 =S for all S ϵ A (X), and ʋ:A(X) → R + with ;ʋ(∅) = 0 . The Shapley-Bondareva-Schmeidler Theorem, which states that a game of the form Γ X (ʋ) = 〈 X, A (X),ʋ〉 is weak if and only if the core of Γ X (ʋ),ζ(Γ X (ʋ)) , is normal, may be regarded as the fundamental theorem for weak cooperative games with side-payments. In this paper we use an ultrapower construction on the reals, R , to summarize a common mathematical theme employed in various constructions used to establish the Shapley-Bondareva-Schmeidler Theorem in the literature (Dalbaen, 1974; Kannai, 1969; Schmeidler, 1967, 1972). This common mathematical theme is that the space L, comprised of finite, real linear combinations of the collection of functions, {χa : a ϵ A (X)}, possesses a certain extension property that is intimately related to the Hahn-Banach Theorem of functional analysis. A close inspection of the extension property reveals that the Shapley-Bondareva-Schmeidler Theorem is in fact equivalent to the Hahn-Banach Theorem.

We work in the context of weak subsystems of second order arithmetic. RCA0 is the system with Δ10 comprehension and Σ10 induction on the natural numbers. WKL0 is RCA0 plus weak Konig's lemma for trees of finite sequences of 0's and 1's. Within RCA0 we encode a separable Banach space  as a countable normed space A over Q . Points of  are Cauchy sequences from A which converge at the rate of at least 2−n. We show that the Hahn-Banach theorem for separable Banach spaces is provably equivalent to WKL0 over RCA0. Thus, once again, WKL0 is revealed as mathematically powerful, despite being proof theoretically equivalent to primitive recursive arithmetic.

Let (Ω, τ, M ) be a nonatomic separable finite measure space. Every continuous functional N on L p ( m ), 1 ⩽ p N ( u + v ) = N ( u ) + N ( v ) whenever uv = 0, is known to be representable by an integral with a nonlinear Caratheodory kernel. Such functionals share several regularity properties with continuous linear functionals. Here we study the question of whether every continuous, disjointly additive functional defined on a closed subspace of L p ( m ) possesses an extension to L p ( m ) with these same properties. This question has applications to the study of nonlinear functionals on Sobolev spaces. It is shown that for a class of subspaces, including those of finite codimension, such an extension always exists, but there are also closed subspaces not possessing this extension property. Analogous results are obtained for disjointly additive mappings from closed subspaces of L p ( m ) into L 1 ( m ) and for functionals defined on subspaces of L ∞ ( m ). The techniques depend heavily on the utilization of Lyapunov vector measures.

This paper considers the problem of the stability robustness computation of polynomials with coefficients which are affine functions of the parameter perturbations. A polynomial is said to be stable if its roots are contained in an arbitrarily pre-specified open set in the complex plane, and its stability robustness is then measured by the norm of the smallest parameter perturbation which destabilizes the polynomial. A simple and numerically effective procedure, which is based on the Hahn-Banach theorem of convex analysis and which is applicable for any arbitrary norm, is obtained to compute the stability robustness. The computation is then further simplified for the case when the norm used is the Holder ∞-norm, 2-norm or 1-norm.

Several properties of non-archimedean weakly closed subspaces in connection with the extension of continuous linear functionals are studied in this paper.

On demontre des theoremes de minimax pour des fonctions prenant des valeurs dans un espace de Riesz. On donne une bonne extension de la version en espace de Riesz du theoreme de Hahn-Banach classique

First we give some elementary properties of the core of a subset relative to a linear subspace. Then we prove a theorem on the frank separation of two convex sets. This theorem admits as particular cases the known theorems on the frank separation and introduces new cases. Finally, we provide a very general version of the Hahn-Banach theorem in an analytic form.