Vector-valued Continuous Functions Research Articles

The Machado–Bishop theorem in the uniform topology

The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of C(I×J,X⊗Y) as the closure of C(I,X)⊗C(J,Y) in different senses and the extension of continuous vector-valued functions are studied.

Norm-multiplicative mappings between spaces of vector-valued continuous functions

Norm-multiplicative mappings between spaces of vector-valued continuous functions

A characterization of nuclear operators on spaces of vector-valued continuous functions with the strict topology

Let X X be a completely regular Hausdorff space, let E E and F F denote Banach spaces. Let C b ( X , E ) C_b(X,E) denote the space of E E -valued bounded continuous functions on X X and let β \beta be the strict topology on this space. We establish the relationship between nuclear operators T : C b ( X , E ) → F T:C_b(X,E)\rightarrow F between the locally convex space ( C b ( X , E ) , β ) (C_b(X,E),\beta ) and the Banach space F F and their representing operator-valued Borel measures.

A numerical range approach to Birkhoff–James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

RESEARCH ON FRACTAL DIMENSIONS AND THE HÖLDER CONTINUITY OF FRACTAL FUNCTIONS UNDER OPERATIONS

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

Vector-Valued Shepard Processes: Approximation with Summability

In this work, vector-valued continuous functions are approximated uniformly on the unit hypercube by Shepard operators. If λ denotes the usual parameter of the Shepard operators and m is the dimension of the hypercube, then our results show that it is possible to obtain a uniform approximation of a continuous vector-valued function by these operators when λ≥m+1. By using three-dimensional parametric plots, we illustrate this uniform approximation for some vector-valued functions. Finally, the influence in approximation by regular summability processes is studied, and their motivation is shown.

On the density of CV0(X)⊗A in CV0(X,A)

Let X be a completely regular Hausdorff space and V a Nachbin family on X. For a locally convex algebra A, let CV0(X,A) be the algebra of all weighted vector-valued continuous functions which vanish at infinity with the topology given by the uniform seminorms induced by V. In this paper we present some sufficient conditions under which CV0(X)⊗A is a dense subspace of CV0(X,A) or isomorphic to it.

English

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon, we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of $\mathcal{C}(K, E)$ spaces.

On some properties of CV_{(0)}(X, A)

Let X be a completely regular Hausdorff space and V a Nachbin familyon X. For a locally convex algebra A, let CV(0)(X, A) be the algebra ofall weighted vector-valued continuous functions with the topology givenby the uniform seminorms induced by V . In this paper we study someproperties of A that are inherited by CV(0)(X, A). These properties arerelated to the unit element, spectral seminorms and uniformly A-convexproperty

Distinguished vector-valued continuous function spaces and injective tensor products

The paper continues the study on the distinguished property of the space $C_{p}\left( X\right) $ (of real-valued continuous functions over a Tychonoff space $X$ in the pointwise topology) under the formation of tensor products towards the following research directions: $\left( i\right) $ the injective tensor product $C_{p}\left( X\right) \otimes _{\varepsilon }E$ of $C_{p}\left( X\right) $ and a real locally convex space $E$ (and its completion $C_{p}\left( X\right) \,\widehat{\otimes }_{\varepsilon }\,E$), and $\left( ii\right) $ the space $C_{p}\left( X,E\right) $ of all $E$-valued continuous functions (with a normed space $E$) endowed with the pointwise topology. This work leads also to a new characterization of distinguished Fréchet locally convex spaces $E$. We show, e.g., that if $C_{p}(X)$ is metrizable, then $E$ is distinguished if and only if metrizable $C_{p}\left( X\right) \otimes _{\varepsilon }E$ is distinguished.

Cauchy Integral and Boundary Value for Vector-Valued Tempered Distributions

Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued analytic functions in tubes TC=Rn+iC obtain vector-valued tempered distributions as boundary values. In a certain vector-valued case, we study the structure of this boundary value, which is shown to be the Fourier transform of the distributional derivative of a vector-valued continuous function of polynomial growth. A set of vector-valued functions used to show the structure of the boundary value is shown to have a one–one and onto relationship with a set of vector-valued distributions, which generalize the Schwartz space DL2′(Rn); the tempered distribution Fourier transform defines the relationship between these two sets. By combining the previously stated results, we obtain a Cauchy integral representation of the vector-valued analytic functions in terms of the boundary value.

Integrable and absolutely continuous vector-valued functions

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual L1 space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.

Ball covering property from commutative function spaces to non-commutative spaces of operators

A Banach space is said to have the ball-covering property (abbreviated BCP) if its unit sphere can be covered by countably many closed, or equivalently, open balls off the origin. Let K be a locally compact Hausdorff space and X be a Banach space. In this paper, we give a topological characterization of BCP, that is, the continuous function space C0(K) has the (uniform) BCP if and only if K has a countable π-basis. Moreover, we give the stability theorem: the vector-valued continuous function space C0(K,X) has the (strong or uniform) BCP if and only if K has a countable π-basis and X has the (strong or uniform) BCP. We also explore more examples for BCP on non-commutative spaces of operators B(X,Y). In particular, these results imply that B(c0), B(ℓ1) and every subspace containing finite rank operators in B(ℓp) for 1<p<∞ all have the BCP, and B(L1[0,1]) fails the BCP. Using those characterizations and results, we show that BCP is not hereditary for 1-complemented subspaces (even for completely 1-complemented subspace in operator space sense) by constructing two different counterexamples.

An example regarding Kalton's paper ‘isomorphisms between spaces of vector-valued continuous functions’

AbstractThe paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_{0}$ which is isomorphic to a closed subspace of a space with a basis and $C(I,\,X)$ is linearly homeomorphic to $C(\Delta ,\, X)$, then $X$ is locally convex, i.e., a Banach space. We will show that Kalton result is sharp by exhibiting non-locally convex quasi Banach spaces $X$ with a basis for which $C(I,\,X)$ and $C(\Delta ,\, X)$ are isomorphic. Our examples are rather specific and actually, in all cases, $X$ is isomorphic to $C(K,\,X)$ if $K$ is a metric compactum of finite covering dimension.

Operators Birkhoff–James Orthogonal to Spaces of Operators

Let be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace we partially solve the operator version of Birkhoff–James orthogonality problem, if is orthogonal to when does there exist a unit vector x 0 such that and is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for is a L 1-predual space as well as a M-ideal in Y, we show that if is orthogonal to then there is a unit vector x 0 with and is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved.

The Mazur–Ulam property on Banach spaces of vector-valued continuous functions

Let K be a totally disconnected locally compact Hausdorff space with |K|≥2, and let X be a strictly convex separable Banach space with dim(XR)≥2. We prove that the Banach space Z=C0(K,X) satisfies the Mazur–Ulam property, namely that every surjective isometry Δ from the unit sphere of Z to the unit sphere of any Banach space Y admits a unique extension to a surjective real-linear isometry from Z to Y. We obtain the following result, if X is a strictly convex Banach space with dim(XR)≥2, then C(C,X) satisfies the Mazur–Ulam property, where C denotes the Cantor set.

Riemann intergability versus continuity for vector-valued functions

The interplay between Riemann integrability and continuity is an interesting topic of modern analysis. In this paper, Riemann integrability of vector-valued continuous functions, property of Lebesgue and weak property of Lebesgue are surveyed and discussed. We also prove that `1(N, X) has the property of Lebesgue.

Orthogonality of sesquilinear forms and spaces of operators

We obtain a necessary and sufficient condition for Birkhoff–James orthogonality in the space of all continuous vector-valued functions from a compact topological space to a finite-dimensional Hilbert space, by using some basic geometric ideas. We apply our outcomes to characterize Birkhoff–James orthogonality of sesquilinear forms. Moreover, using Birkhoff–James orthogonality of sesquilinear forms, we obtain a different proof of the renowned Bhatia–Šemrl Theorem.

Conditions of unicity of the best nonsymmetric $L_1$-approximant for vector-valued functions by one-dimensional subspace

The conditions of the unicity of the best nonsymmetric $L_1$-approximant for the continuous vector-valued functions on the metric compact by one-dimensional subspace are obtained.

Orthogonality of bilinear forms and application to matrices

We characterize Birkhoff-James orthogonality of continuous vector-valued functions on a compact topological space. As an application of our investigation, Birkhoff-James orthogonality of real bilinear forms are studied. This allows us to present an elementary proof of the well-known Bhatia-Šemrl Theorem in the real case.