Spaces Of Continuous Vector-valued Functions Research Articles

Local Uniform Strong Proximinality of the Space of Continuous Vector-Valued Functions

ABSTRACTLet X be a Banach space, S be a compact Hausdorff space and Y be a U-proximinal subspace of X. We prove that C(S,Y) is locally uniformly strongly proximinal in C(S,X) and the corresponding metric projection map is Hausdorff metric continuous.

Korovkin theory for vector-valued functions on a locally compact space

We consider ordered spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with appropriate locally convex topologies. Using suitable sets of such functions as test systems a Korovkin type approximation theorem for equicontinuous nets of positive operators is established. As in the classical theory, the Korovkin closure is characterized both through envelopes of functions and through measure theoretical conditions.

Some aspects of the theory of integro-differential Barbashin equations in function spaces

We study the solvability conditions for the Cauchy problem for the Barbashin integro-differential equation in the space of bounded continuous functions and in the space of continuous vector-valued functions with the values in an ideal Banach space. We also consider conditions guaranteeing the existence of classical and generalized solutions to the Cauchy problem for systems of linear Barbashin integro-differential equations. Bibliography: 5 titles.

Spaces of vector-valued functions and their duals

We consider spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with classes of locally convex topologies, which include and generalize various known ones such as weighted space- or inductive limit-type topologies. The main result states that every continuous linear functional on such a function space can be expressed as an integral with respect to some canonical (dual space-valued) vector measure.

Extreme points and isometries on vector-valued Lipschitz spaces

For a Banach space E and a compact metric space ( X , d ) , a function F : X → E is a Lipschitz function if there exists k > 0 such that ‖ F ( x ) − F ( y ) ‖ ⩽ k d ( x , y ) for all x , y ∈ X . The smallest such k is called the Lipschitz constant L ( F ) for F. The space Lip ( X , E ) of all Lipschitz functions from X to E is a Banach space under the norm defined by ‖ F ‖ = max { L ( F ) , ‖ F ‖ ∞ } , where ‖ F ‖ ∞ = sup { ‖ F ( x ) ‖ : x ∈ X } . Recent results characterizing isometries on these vector-valued Lipschitz spaces require the Banach space E to be strictly convex. We investigate the nature of the extreme points of the dual ball for Lip ( X , E ) and use the information to describe the surjective isometries on Lip ( X , E ) under certain conditions on E, where E is not assumed to be strictly convex. We make use of an embedding of Lip ( X , E ) into a space of continuous vector-valued functions on a certain compact set.

Banach-Stone theorems for Banach bundles

We prove two Banach-Stone type theorems for section spaces of real Banach bundles. The first theorem assumes that the duals of all fibers are strictly convex, and the second considers disjointness-preserving operators. In each case, the result generalizes the corresponding Banach-Stone theorem for spaces of continuous vector-valued functions.

On Subspaces and Quotients of Banach Spaces C(K, X)

We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces.

Antiproximinal Sets in Banach Spaces of Continuous Vector-Valued Functions

A closed nonvoid subset Z of a Banach space X is called antiproximinal if no point outside Z has a nearest point in Z. The aim of the present paper is to prove that, for a compact Hausdorff space T and a real Banach space E, the Banach space C(T,E), of all continuous functions defined on T and with values in E, contains an antiproximinal bounded closed convex body. This extends a result proved by V. S. Balaganskii (1996, Mat. Zametki60, 643–657) in the case E=R.

Perturbation of a convex-valued operator by a set-valued map of Hammerstein type with non-convex values, and boundary-value problems for functional-differential inclusions

A functional inclusion in the space of continuous vector-valued functions on the interval [a,b] is considered, the right-hand side of which is the sum of a convex-valued set-valued map and the product of a linear integral operator and a set-valued map with images convex with respect to switching. Estimates for the distance between a solution of this inclusion and a fixed continuous vector-valued function are obtained and the structure of the set of solutions of this inclusion is studied on the basis of these estimates. A result on the density of the solutions of this inclusion in the set of solutions of the 'convexized' inclusion is obtained and the 'bang-bang' principle for the original inclusion is established. This theory is applied to the study of the solution sets of boundary-value problems for functional-differential inclusions with non-convex right-hand sides.

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

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

Fréchet spaces of continuous vector-valued functions: Complementability in dual Fréchet spaces and injectivity

Fréchet spaces of continuous vector-valued functions: Complementability in dual Fréchet spaces and injectivity

Nuclear operators on spaces of continuous vector-valued functions

Let Ω: be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of continuous E-valued functions on Ω under supnorm. It is well known [3, p. 182] that if F is a Banach space then any bounded linear operator T:C(Ω, E)→ F has a finitely additive vector measure G defined on the σ-field of Borel subsets of Ω with values in the space ℒ(E, F**) of bounded linear operators from E to the second dual F** of F. The measure G is said to represent T. The purpose of this note is to study the interplay between certain properties of the operator T and properties of the representing measure G. Precisely, one of our goals is to study when one can characterize nuclear operators in terms of their representing measures. This is of course motivated by a well-known theorem of L. Schwartz [5] (see also [3, p. 173]) concerning nuclear operators on spaces C(Ω) of continuous scalar-valued functions. The study of nuclear operators on spaces C(Ω, E) of continuous vector-valued functions was initiated in [1], where the author extended Schwartz's result in case E* has the Radon-Nikodym property. In this paper, we will show that the condition on E* to have the Radon-Nikodym property is necessary to have a Schwartz's type theorem. This leads to a new characterization of dual spaces E* with the Radon-Nikodym property. In [2], it was shown that if T:C(Ω, E)→ F is nuclear than its representing measure G takes its values in the space (E, F) of nuclear operators from E to F. One of the results of this paper is that if T:C(Ω, E)→ F is nuclear then its representing measure G is countably additive and of bounded variation as a vector measure taking its values in (E, F) equipped with the nuclear norm. Finally, we show by easy examples that the above mentioned conditions on the representing measure G do not characterize nuclear operators on C(Ω, E) spaces, and we also look at cases where nuclear operators are indeed characterized by the above two conditions. For all undefined notions and terminologies, we refer the reader to [3].

Integral operators on spaces of continuous vector-valued functions

Let X be a compact Hausdorff space, let E be a Banach space, and let C(X,E) stand for the Banach space of E-valued continuous functions on X under the uniform norm. In this paper we characterize integral operators (in the sense of Grothendieck) on C(X,E) spaces in terms of their representing vector measures. This is then used to give some applications to nuclear operators on C(X,E) spaces

Integral Operators on Spaces of Continuous Vector-Valued Functions

Let $X$ be a compact Hausdorff space, let $E$ be a Banach space, and let $C(X,E)$ stand for the Banach space of $E$-valued continuous functions on $X$ under the uniform norm. In this paper we characterize integral operators (in the sense of Grothendieck) on $C(X,E)$ spaces in terms of their representing vector measures. This is then used to give some applications to nuclear operators on $C(X,E)$ spaces.

ON THE INDEX AND SPECTRUM OF INTEGRAL OPERATORS OF POTENTIAL TYPE ALONG RADON CURVES

A study is made of how classical integral equations of mathematical physics are affected by nonregularity of the contour of integration. A criterion is obtained for a matrix integral equation with operator of potential type acting in to be Noetherian, and the index is computed. It is established that an integral equation corresponding to the interior Dirichlet problem for harmonic functions is Noetherian in for all except for a finite or countable number of values determined by the angles of the contour; the defect numbers, which depend on and the angles mentioned, are found. Analogous results are obtained for the system of integral equations of the planar theory of elasticity. The non-Noetherian spectrum of a matrix integral operator of potential type acting in a space of continuous vector-valued functions is described. This result is illustrated by an example of an operator in elasticity theory (for which, in particular, the Fredholm radius is found) and of the direct value of a double layer potential.

Best approximation in the space of continuous vector-valued functions

Using a well-known characterization theorem for best approximations, direct proofs are given of some (generalizations of) recent results of Tanimoto who deduced them from a general minimax theorem that he first established.

Spaces of Continuous Vector Functions as Duals

AbstractA well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.

On approximation in weighted spaces of continuous vector-valued functions

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

Integral representation of multiplicative, involution preserving operators in 𝔏(ℭ₀(𝔖,𝔄),𝔅)

Bounded linear operators from the space of continuous vector-valued functions which preserve multiplication and involution are characterized in terms of their representing measures. A key role is played by the Arens product in the second dual of a Banach algebra.

Approximation by certain subspaces in the banach space of continuous vector-valued functions

A theory of best approximation is developed in the normed linear space C( T, E), the space of E-valued bounded continuous functions on the locally compact Hausdorff space T, with the supremum norm. The approximating functions belong to the subspace C F ( T, E) of C( T, E) consisting of those functions which have “limit at infinity” which lies in the subspace F of the normed linear space E. A distance formula is obtained, and a selection for the metric projection onto C f ( T, E) is constructed which has many desirable properties. The theory includes study of best approximation in l ∞ by the subspace c 0, and closely parallels the known theory of best approximation by M-ideals (although our subspace is not an M-ideal, in general).