Topologies on spaces of vector-valued continuous functions

Extension of operators on spaces of vector-valued continuous functions

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.

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let E be a locally convex Hausdorff space. Let CV 0 ( X, E ) and CV b ( X, E ) be the weighted locally convex spaces of vector‐valued continuous functions with a topology generated by seminorms which are weighted analogue of the supremum norm. In the present paper, we characterize multiplication operators and weighted composition operators on the spaces CV 0 ( X, E ) and CV b ( X, E ) induced by scalar‐valued and vector‐valued mappings. A (linear) dynamical system on these weighted spaces is obtained as an application of the theory of multiplication operators.

Let 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.

A. Pietsch has shown that the class of dominated linear operators on C(5) coincides with the class of absolutely summing operators.If the space C(S) is replaced by CX(S), where A" is a Banach space, this is no longer the case.However, any absolutely summing operator is always dominated, and the classes of operators coincide exactly when X is finite dimensional.A characterization of absolutely summing operators on CX(S) is given.

Local symmetry and smoothness in the space of vector-valued continuous functions

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.

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 the universal completion of pointfree function spaces

Antiproximinal Sets in Banach Spaces of Continuous Vector-Valued Functions

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

On continuous selections for metric projections in spaces of continuous functions

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.

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

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.