Set-Valued Mappings on LCA Group and Approximate Identity

A family \(\mathcal {F}\) of sets is said to satisfy the (p, q)-property if among any p sets of \(\mathcal {F}\) some q have a non-empty intersection. The celebrated (p, q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in \(\mathbb {R}^d\) that satisfies the (p, q)-property for some \(q \ge d+1\), can be pierced by a fixed number (independent of the size of the family) \(f_d(p,q)\) of points. The minimum such piercing number is denoted by \(\mathsf {HD} _d(p,q)\). Already in 1957, Hadwiger and Debrunner showed that whenever \(q>\frac{d-1}{d}\,p+1\) the piercing number is \(\mathsf {HD} _d(p,q)=p-q+1\); no tight bounds on \(\mathsf {HD} _d(p,q)\) were found ever since. While for an arbitrary family of compact convex sets in \(\mathbb {R}^d\), \(d \ge 2\), a (p, 2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific classes. The best-studied among them is the class of axis-parallel boxes in \(\mathbb {R}^d\), and specifically, axis-parallel rectangles in the plane. Wegner (Israel J Math 3:187–198, 1965) and (independently) Dol’nikov (Sibirsk Mat Ž 13(6):1272–1283, 1972) used a (p, 2)-theorem for axis-parallel rectangles to show that \(\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1\) holds for all \(q \ge \sqrt{2p}\). These are the only values of q for which \(\mathsf {HD} _\mathrm{{rect}}(p,q)\) is known exactly. In this paper we present a general method which allows using a (p, 2)-theorem as a bootstrapping to obtain a tight (p, q)-theorem, for classes with Helly number 2, even without assuming that the sets in the class are convex or compact. To demonstrate the strength of this method, we show that \(\mathsf {HD} _{d\text {-box}}(p,q)=p-q+1\) holds for all \(q > c' \log ^{d-1} p\), and in particular, \(\mathsf {HD} _\mathrm{{rect}}(p,q)=p-q+1\) holds for all \(q \ge 7 \log _2 p\) (compared to \(q \ge \sqrt{2p}\), obtained by Wegner and Dol’nikov more than 40 years ago). In addition, for several classes, we present improved (p, 2)-theorems, some of which can be used as a bootstrapping to obtain tight (p, q)-theorems. In particular, we show that any class \(\mathcal {G}\) of compact convex sets in \(\mathbb {R}^d\) with Helly number 2 admits a (p, 2)-theorem with piercing number \(O(p^{2d-1})\), and thus, satisfies \(\mathrm {HD}_{\mathcal {G}}(p,q) = p-q+1\), for a universal constant c.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

A Helly-type theorem for higher-dimensional transversals

LetK be a family of compact convex sets in the plane. We show that if every three members ofK admit a common line transversal, then there exist four lines which together meet all the members ofK.

This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous set-valued functions and in spaces of vector valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions.

Derivation of necessary conditions for optimality in optimal control theory relies on variational calculus of control systems. We provide here such calculus for the semilinear differential inclusion $$x' \in Ax + F(t,x), x(0) = x_0$$ (1) (1) where A is the infinitesimal generator of a C o-semigroup on a separable Banach space X and F: [0,T]×X∼ X is a set-valued map.

It is shown that the problem of the best uniform approximation in the Hausdorff metric of a continuous set-valued map with finite-dimensional compact convex images by constant set-valued maps whose images are balls in some norm can be reduced to a visual geometric problem. The latter consists in constructing a spherical layer of minimal thickness which contains the complement of a compact convex set to a larger compact convex set.

Using nonstandard analysis, a new method is introduced for locally compact Abelian topological groups. Hyperfiniteness plays an important role. All LCA groups can be obtained this way. This yields approximation results for the (generalized) Fourier transform.

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

Measure differential inclusions were introduced by J. J. Moreau to study sweeping processes, and have since been used to study rigid body dynamics and impulsive control problems. The basic formulation of an MDI is “d μ / d ν(t) ∈ K(t)” where μ is a vector measure, ν an unsigned measure, and K(⋅) is a set-valued map with closed, convex values and is hemicontinuous. Note that μ need not be absolutely continuous with respect to ν. Stewart extended Moreau's original concept (which applied only to cone-valued K(⋅)) to general convex sets, and gave strong and weak formulations of “d μ / d ν(t) ∈ K(t)” where K(t) ⊂ Rn. Here the strong and weak formulations of Stewart are extended to infinite-dimensional problems where K(t) ⊂ X where X is a separable reflexive Banach space; they are shown to be equivalent under mild assumptions on K(⋅).

For an arbitrary normed algebra with an approximate identity, we introduce some notions of invertibility of a net in this algebra with respect to the given approximate identity. By means of that, we show a Wiener-Tauberian type theorem in that general situation. An application yields an abstract Wiener-Tauberian theorem for a class of commutative Banach algebras, which implies the classical Wiener-Tauberian theorem for locally compact Abelian groups.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

Let X X be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for X X in terms of Lipschitz retractions onto generating compact subsets K K of X X . In one direction, if X X admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set K K . On the other hand, we prove that if X X admits a “small” generating compact Lipschitz retract then X X has the π \pi -property. It is still unknown if the π \pi -property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces X X . We give an example of a small generating convex compact set which is not a Lipschitz retract of C [ 0 , 1 ] C[0,1] , although it is contained in a small convex Lipschitz retract and contains another one. We characterize isomorphically Hilbertian spaces as those Banach spaces X X for which every convex and compact subset is a Lipschitz retract of X X . Finally, we prove that a convex and compact set K K in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map.

UDC 513.88 It the article we study sublinear functionals that are defined on the space of Bochner integrable functions and possess the properties of decomposition and scalar compactness. We give an integral representation for the functionals. Such functionals are used in proving theorems on existence of continuous selections for a family of lower semicontinuous multivalued mappings with decomposable closed nonconvex values in the space of Bochner integrable functions. The article relates to the studies [1-3] and others. 1. We present basic definitions, notation, and a series of auxiliary results. Throughout what follows, X is a separable Banach space with norm H" ]l, T is a Hausdorff compact space with positive Radon measure/z and a-algebra ~ of/,-measurable subsets, and R stands for the real axis. We denote by L1 (T, X) the Banach space of equivalence classes of Bochner integrable functions u: T ---* X, with the norm I[u(.)HL1 = fJ[u(t)Hd#. T By X t and L ~ (T, X) we mean the topological dual spaces of X and L1 (T, X) with the strong (norm) topologies; (x, xl~ and (u(-), u'(.)) are the canonical bilinear forms establishing duality between X and X' and between LI(T, X) and UI(T , X) respectively. For the norm of X ~ we use the same notation as for that of X. The spaces X' and LI(T,X ) endowed with weak a(X~,X) - and a( L~ (T,X), Ll(T,X) )-topologies will be denoted by a-X' and a-L](T,X) (see [4, p. 6831). A function v : T ~ X I is called scalarly measurable if, given x E X, the scalar function (x, v(t)) is measurable. A function v : T ~ X ~ is called weakly measurable if, for every e > 0, there is a closed set Te C T, p(T \ Te) <_ e, such that the restriction of v to Te is a continuous function from T~ into a-X ~. We denote by Loo(T, X') the Banach space of equivalence classes of scalarly measurable, essentially bounded functions v : T --* X', with the norm IIv(.)Hoo = esssup Hv(t)H. Henceforth we identify lET equivalence classes in the spaces L~ (T, X) and L~ (T, X') with their representatives. It is well known (see [4, p. 807]) that U I(T,X) and Loo(T,X') are linearly isometric and that each element x'(.) e L~(T,X) is representable as (x(.),x'(.)) = f(x(t),v(t))d#, x(.) �9 LI(T,X), where T v(.) �9 Loo(T,X'). Therefore, we shall by convention regard Loo(T,X ~) as strongly dual to LI(T, X) and write (x(.),x~(.)) = f(x(t),x~(t))d#, x(.) �9 LI(T,X), considering x~(.) as an element of the space

In a recent paper, F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E), the Banach space of continuous functions from a compact Hausdorff space Q to E, has the Dieudonne property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of l1. In this paper we give a positive answer to their question. As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonne property if and only if C(Q, E) has the same property. If E is a Banach space and Q is a compact Hausdorff space, then C(Q, E) will stand for the Banach space of the E-valued continuous functions on Q under the supremum norm. A Banach space E is said to have the Dieudonne property if for every Banach space F, any bounded linear operator T: E -> F that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. In [3] F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E) has the Dieudonne property and they asked whether the same result is true when replacing the assumption that E* is separable by supposing only that 11 does not embed in E. In this paper we give a positive answer to their question. Recall that a topological space (X, -y) is said to be Polish if it is homeomorphic to a separable complete metric space and it is said to be analytic if it is the continuous image of a Polish space. A subset A of a topological space (X, y) is said to be coanalytic if its complement (X\A, y) is analytic. Finally A is said to be PCA if it is the continuous image of a coanalytic space. The notations and terminology used and not defined can be found in [5, 8, or 10]. In the proof of Lemma 3 we need the following two results. THEOREM 1 (M. SREBRNY [9]). Let X and Y be two analytic spaces and let F be a multivalued function from X to the subsets of Y, such that its graph is PCA and for which one can prove that for every x c X, F(x) :8 0 using only the axioms of ZFC. Then there exists a universally measurable map f: X -> Y such that f (x) c F(x) for every x E X. THEOREM 2 (I. ASSANI [1, 2]). Let E be a separable Banach space. The set of weakly Cauchy sequences is a coanalytic subset of EN. Received by the editors January 5, 1985. 1980 Mathematics Subject Classification. Primary 46G10, 46B22.