Vector-valued Continuous Functions Research Articles

Linear isometries between certain subspaces of continuous vector-valued functions

Linear isometries between certain subspaces of continuous vector-valued functions

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.

Compact and weakly compact weighted composition operators on weighted spaces of continuous functions

In this note we characterize the compact and weakly compact weighted composition operatorsWπ,ϕ on certain weighted locally convex spacesCVo(X, E) of vector-valued continuous functions induced by self maps ϕ ofX and the operator-valued mappings π ∶X→B(E).

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

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

Weak Sequential Convergence and Weak Compactness in Spaces of Vector-Valued Continuous Functions

Let X be completely regular Hausdorff space, E a Hausdorff locally convex space, C(X, E) (Cb(X, E)) the space of all E-valued (all E-valued bounded) continuous functions on X, and βz a strict topology on Cb(X, E). It is proved that a sequence {ƒn} in (Cb(X, E), βz) converges weakly to 0 if and only if {ƒn} is uniformly bounded and ƒn,(x) → 0, pointwise on X with weak topology on E. Similar results are proved for (C(X, E), β∞c). Some characterizations of relatively weakly compact subsets of (Cb(X, E), βz) and (C(X, E), β∞c) are also given.

The Convex Hull of Extremal Vector-Valued Continuous Functions

Let T be a completely regular space and X a strictly convex n-dimensional real space. We prove that every continuous function from T into the closed unit ball of X can be expressed as an average of eight continuous functions from T into the sphere of X if and only if dim (T) ⩽ n−1, where dim(T) denotes the covering dimension of T. The proof we give can be used to prove the same fact, without hypotheses on T, when X is infinite-dimensional, although in this case it has been proved recently that a better result can be obtained.

Bornological spaces of null sequences

It is proved that c0(E) is an LB-space, whenever E is either a Montel LB-space or a coechelon space k∞(v) of type ∞ (a partial solution of a problem of Schmets and Bierstedt). In the second case we show that c0(k∞(v)) is even the Mackey completion of c0 ⊗e k∞(v). Introduction. J. Schmets has successfully studied various linear-topological properties of spaces of vector-valued continuous functions [Sch2]. Unfortunately, the problem of bornologicity of such spaces has been left open (comp. [Sch1]). Up to now a solution has been obtained only in very special cases (see [Bo1], [Mu], [Sch2, I.7.3 and IV.4.7], [DG], [BoS1] and [BoS2]), in particular, it is still unclear when c0(E) is bornological. Bierstedt and Schmets asked if c0(E), E a DFM-space, is bornological (or, equivalently, an LB-space). We solve this problem completely. Then we show that the problem of bornologicity of c0(E) for general LB-spaces E (also asked by Schmets and Bierstedt [Sch1], see also [Sch2, Ch. IV]) reduces in some sense to the same problem for Montel E. By this reduction we solve the problem for arbitrary coechelon spaces E = k∞(v). We use our “factorization” approach developed in [DiDo1] and [DiDo2]. The problem of boronologicity of C(K,E), K compact, at least in case when E = indn∈N En is an LBor LF-space, is closely related to the question of interchanging of e-products and inductive limits (comp. [BoS2]). Indeed, by a result of Mujica ([Mu], [Sch2, I.7.2]), it is known that C(K,E) = C(K)e indn∈N En contains a bornological space indn∈N(C(K)eEn) as a topological subspace. Thus, if (∗) C(K)e ind n∈N En = ind n∈N (C(K)eEn) holds algebraically, then it holds topologically and C(K,E) is a fortiori bornological (see the result of Mujica [Sch2, I.7.3] or Marquina and Schmets [MS] and a more general result of Defant and Govaerts [Sch2, IV.4.7], [DG, Th. 13, Cor. 14]). If the space E is compactly regular (i.e., each compact subset of E is compact in some step En), then (∗) holds. Roughly speaking, for LB-spaces the case of 1991 Mathematics Subject Classification. Primary 46A11, 46A13, 46E40.

Automatic continuity of homomorphisms and derivations on algebras of continuous vector-valued functions

Automatic continuity of homomorphisms and derivations on algebras of continuous vector-valued functions

Approximation and extension of continuous functions

AbstractIn this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

A reduction principle for topological classification of nonautonomous differential equations

SynopsisThe paper is concerned with equations of the form x' = A(t)x +f(t, x), where A is a continuous matrix function defined on ℝ, f is a continuous vector-valued function of (t, x) with f(t, 0) = 0. It is proved that if x' = A(t)x has an exponential trichotomy, A is bounded and f satisfies the Lipschitz condition with coefficient sufficiently small, then these equations are topologically equivalent to the systems of equations of the form , where B, g satisfy the same conditions as A, f.

Multipliers in continuous vector-valued function spaces

We identify the multiplier space of continuous vector-valued function spaces as the bounded continuous multiplier-valued space in the strong topology.

Multiplication Operators and Dynamical Systems

AbstractLet X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CVb(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CVb(X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.

P-adic Ascoli theorems and compactoid polynomials

We prove several properties of compactoid sets in some spaces of vector-valued continuous functions. As a consequence we derive some p-adic Ascoli theorems which allow us to answer the question proposed by W.H. Schikhof in [MR.90i 46128].

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.

Multiplication operators on weighted spaces of vector-valued continuous functions

AbstractIf V is a system of weights on a completely regular Hausdorff space X and E is alocally convex space, then CV0(X, E) and CVb (X, E) are locally convex spaces of vector-valued continuous functions with topologies generated by seminorms which are weighted analogues of the supremum norm. In this paper we characterise multiplication operators on these spaces induced by scalar-valued and vector-valued mappings. Many examples are presented to illustrate the theory.

On weak approximation and convexification in weighted spaces of vector-valued continuous functions

Let X be a completely regular Hausdorff space. A Nachbin family of weights is a set V of upper-semicontinuous positive functions on X such that if u, υ ∈ V then there exists w ∈ V and t > 0 so that u, υ ≤ tw. For any Hausdorff topological vector space E, the weighted space CV0(X, E) is the space of all E-valued continuous functions f on X such that υf vanishes at infinity for all υ ∈ V. CV0(X, E) is equipped with the weighted topologywv = wv(X, E) which has as a base of neighbourhoods of zero the family of all sets of the formwhere υ ∈ Vand W is a neighbourhood of zero in E. If E is the scalar field, then the space CV0(X, E) is denoted by CV0(X). The reader is referred to [4, 6, 8] for information on weighted spaces.