Antiproximinal Sets in Banach Spaces of Continuous Vector-Valued Functions

In this note an analytic representation is given for continuous linear operators from C ( X ) C(X) into a linear normed space Y where C ( X ) C(X) is the space of continuous functions on [0, 1] with values in a linear normed space X.

On Baire-Like Spaces of Spaces of Continuous Vector-Valued Functions

A Holsztyński theorem for spaces of continuous vector-valued functions

The strict, superstrict and the F topologies are defined on a space A of continuous functions from a completely regular space into a Banach space E. Properties of these topologies are discussed and the corresponding dual spaces are identified with certain spaces of operator-valued measures.In case E is a Banach lattice, A becomes a lattice under the pointwise ordering and the strict and superstrict duals of A coincide with the spaces of all r-additive and all o-additive functional on A respectively.Introduction.The Riesz representation theorem says that any continuous linear functional F on the space of continuous real functions on a compact Hausdorff space X with the uniform topology must have the form Fif) = fxf dm for some bounded regular Borel measure on X.This representation was extended later to other spaces, first in case X is locally compact and later by Aleksandrov [1] for continuous linear functionals on the space Cb{X) of all bounded continuous real functions on a completely regular space.The representation was given by means of integrals with respect to members of the space M(X) of all bounded, finitely-additive, regular with respect to zero sets, measures on the algebra generated by the zero sets.The o-additive, r-additive and tight linear functionals correspond to the o-additive, r-additive and tight members of M(X) respectively (see Varadarajan [24]).Buck [4], for the locally compact case, and Sentilles [23], for the completely regular case, have defined the strict topologies on Cb(X) which yield as dual spaces certain subspaces of M(X).Several others like Hewitt [10], Bogdanowich [2], Wells [25], the author [12] and others have considered the problem of representation of linear functionals on spaces of continuous scalarvalued or vector-valued functions.In this paper we define certain locally convex topologies on spaces of continuous vector-valued functions on a completely regular space.We study some of the properties of these topologies and represent their duals with operator-valued measures on certain o-algebras of subsets of X.The

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.

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 K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm. We study in this paper Banach-Saks operators defined on C(K, E) spaces. We characterize these operators for a large class of compacts K (the scattered ones), or for a large class of Banach spaces E (the superreflexive ones). We also show by some examples that our theorems can not be extended directly.

This paper is concerned with a class of problem of best simultaneous approximations from $RS$-sets in Banach spaces $X$. It is shown that the best simultaneous approximations from an $RS$-set is strongly unique in the case when $X$ is a real Banach space, and strongly unique of order $\alpha \geq 2$ in the case when $X$ is a complex Banach space.

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.

An Application of M-Structure to Theorems of the Banach-Stone Type

Extension of operators on spaces of vector-valued continuous functions

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on , there exists a linear subspace of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = exp n +1τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ .

The set St(f1, f2, f3) of Steiner points is described for any three functions f1, f2, f3 in the space C[K] of real-valued continuous functions on a Hausdorff compact set K. The set St(f1, f2, f3) consists of all functions s ∈ C[K] such that the sum ‖f1 − s‖ + ‖f2 − s‖ + ‖f3 − s‖ is minimal. It is proved that the set St(f1, f2, f3) is not empty; the triples f1, f2, f3 having a unique Steiner point are described; a Lipschitz selection is presented for the mapping (f1, f2, f3) → St(f1, f2, f3). These results imply the description of all real two-dimensional Banach spaces possessing the following property: the sum ‖x1 − s‖ + x2 − s‖ + ‖x3 − s‖ is equal to the semiperimeter of the triangle x1x2x3 for any triple x1, x2, x3 and some of its Steiner point s = s(x1, x2, x3).

A new class of generalized set-valued implicit variational inclusions in banach spaces with an application

En este trabajo probamos existencia y unicidad de las soluciones para una ecuacion de Volterra, con condicion inicial, en el espacio de funciones continuas con variacion acotada y valores en un espacio de Banach arbitrario. Damos una formula de variacion de parametros para las soluciones de cierta clase de ecuacion lineal integral. Finalmente probamos la controlabilidad exacta de una ecuacion integral particular usando esa formula. Mas aun, bajo cierta condicion, encontramos una formula para una direccion de control de un tipo de Sistema que se estudia en el presente trabajo, desde un estado inicial a uno final en un tiempo prescrito.