UNIQUENESS OF THE TOPOLOGY ON SPACES OF VECTOR-VALUED FUNCTIONS

2001 ◽  
Vol 64 (2) ◽  
pp. 445-456 ◽  
Author(s):  
A. R. VILLENA
Keyword(s):  
Banach Space ◽  
Linear Subspace ◽  
Locally Compact Group ◽  
Translation Invariant ◽  
Linear Spaces ◽  
Space Topology ◽  
Isolated Points ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Topological Linear

Let Ω be a topological space without isolated points, let E be a topological linear space which is continuously embedded into a product of countably boundedly generated topological linear spaces, and let X be a linear subspace of C(Ω, E). If a ∈ C(Ω) is not constant on any open subset of Ω and aX ⊂ X, then it is shown that there is at most one F-space topology on X that makes the multiplication by a continuous. Furthermore, if [Ufr ] is a subset of C(Ω) which separates strongly the points of Ω and [Ufr ]X ⊂ X, then it is proved that there is at most one F-space topology on X that makes the multiplication by a continuous for each a ∈ [Ufr ].These results are applied to the study of the uniqueness of the F-space topology and the continuity of translation invariant operators on the Banach space L1(G, E) for a noncompact locally compact group G and a Banach space E. Furthermore, the problems of the uniqueness of the F-algebra topology and the continuity of epimorphisms and derivations on F-algebras and some algebras of vector-valued functions are considered.

A Choquet theorem for general subspaces of vector-valued functions

1985 ◽  
Vol 98 (2) ◽  
pp. 323-326 ◽  
Author(s):  
Paulette Saab ◽  
Michel Talagrand
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Vector Measure ◽  
Complex Banach Space ◽  
Supremum Norm ◽  
Bounded Linear ◽  
Vector Valued Functions ◽  
Bounded Linear Functional ◽  
Vector Valued

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

scholarly journals POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

On Learning Vector-Valued Functions

2005 ◽  
Vol 17 (1) ◽  
pp. 177-204 ◽  
Author(s):  
Charles A. Micchelli ◽  
Massimiliano Pontil
Keyword(s):  
Hilbert Space ◽  
Learning Theory ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Support Vector ◽  
Translation Invariant ◽  
Minimal Norm ◽  
Finite Set ◽  
Vector Valued Functions ◽  
Vector Valued

In this letter, we provide a study of learning in a Hilbert space of vector-valued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.

scholarly journals Abelian ergodic theorems for vector-valued functions

1975 ◽  
Vol 16 (1) ◽  
pp. 57-60 ◽  
Author(s):  
P. E. Kopp
Keyword(s):  
Banach Space ◽  
Ergodic Theorem ◽  
Convergence Theorem ◽  
Direct Application ◽  
Approximation Technique ◽  
Ergodic Theorems ◽  
Resolvent Equation ◽  
Continuous Parameter ◽  
Vector Valued Functions ◽  
Vector Valued

This note contains extensions of the Abelian ergodic theorems in [3] and [6] to functions which take their values in a Banach space. The results are based on an adaptation of Rota's maximal ergodic theorem for Abel limits [8]. Convergence theorems for continuous parameter semigroups are deduced by the approximation technique developed in [3], [6]. A direct application of the resolvent equation also enables us to deduce a convergence theorem for pseudo-resolvents.

The spectrum and eigenspaces of a meromorphic operator-valued function

1997 ◽  
Vol 127 (5) ◽  
pp. 1027-1051 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Spectral Theory ◽  
Approximation Theorem ◽  
Meromorphic Mapping ◽  
Bounded Operators ◽  
Single Operator ◽  
Definition Of ◽  
Vector Valued Functions ◽  
Vector Valued

SynopsisIt is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

scholarly journals Nuclear operators on spaces of continuous vector-valued functions

1991 ◽  
Vol 33 (2) ◽  
pp. 223-230 ◽  
Author(s):  
Paulette Saab ◽  
Brenda Smith
Keyword(s):  
Banach Space ◽  
Vector Measure ◽  
Type Theorem ◽  
Representing Measure ◽  
Bounded Linear ◽  
Continuous Vector ◽  
Nuclear Operators ◽  
Nikodym Property ◽  
Vector Valued Functions ◽  
Vector Valued

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].

scholarly journals SCALAR BOUNDEDNESS OF VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 325-333 ◽  
Author(s):  
MATÍAS RAJA ◽  
JOSÉ RODRÍGUEZ
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Sufficient Conditions ◽  
Vector Integration ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractWe provide sufficient conditions for a Banach space-valued function to be scalarly bounded, which do not require to test on the whole dual space. Some applications in vector integration are also given.

Rotundity In Köthe Spaces of Vector-Valued Functions

1989 ◽  
Vol 41 (4) ◽  
pp. 659-675 ◽  
Author(s):  
A. Kamińska ◽  
B. Turett
Keyword(s):  
Banach Space ◽  
Analogous Result ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Uniform Rotundity ◽  
Köthe Spaces ◽  
Bochner Space ◽  
Köthe Space ◽  
Vector Valued Functions ◽  
Vector Valued

In this paper, Köthe spaces of vector-valued functions are considered. These spaces, which are generalizations of both the Lebesgue-Bochner and Orlicz-Bochner spaces, have been studied by several people (e.g., see [1], [8]). Perhaps the earliest paper concerning the rotundity of such Köthe space is due to I. Halperin [8]. In his paper, Halperin proved that the function spaces E(X) is uniformly rotund exactly when both the Köthe space E and the Banach space X are uniformly rotund; this generalized the analogous result, due to M. M. Day [4], concerning Lebesgue-Bochner spaces. In [20], M. Smith and B. Turett showed that many properties akin to uniform rotundity lift from X to the Lebesgue-Bochner space LP(X) when 1 < p < ∞. A survey of rotundity notions in Lebesgue-Bochner function and sequence spaces can be found in [19].

Multipliers for vector valued functions

1984 ◽  
Vol 99 (1-2) ◽  
pp. 137-143
Author(s):  
José Luis Torrea
Keyword(s):  
Abelian Group ◽  
Banach Spaces ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
Translation Invariant ◽  
Bounded Linear ◽  
Vector Valued Functions ◽  
Vector Valued

SynopsisLet G be a locally compact abelian group and let Γ be the dual of G. Let A, B be Banach spaces and Lp(G,A) the Bochner-Lebesgue spaces. We prove that the space of bounded linear translation invariant operators from L1(G, A) to LX(G, B) can be identified with the space of bounded convolution invariant (in some sense) operators and also with the space of a(A, B)-valued “weak regular” measures with the relation Tf = f *μ. (A. The existence of a function m∈ L∞ (Γ,α(A,B)), such that is also proved.

scholarly journals Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

2010 ◽  
Vol 18 (4) ◽  
pp. 207-212 ◽  
Author(s):  
Takao Inoué ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Vector Valued Functions ◽  
Vector Valued

Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).

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