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On vector-valued functions and the ε-product
Citation Link: https://doi.org/10.15480/882.4898
Publikationstyp
Doctoral Thesis
Date Issued
2023
Sprache
English
Author(s)
Title Granting Institution
Technische Universität Hamburg
Place of Title Granting Institution
Hamburg
Examination Date
2022-07-01
Institut
TORE-DOI
This habilitation thesis centres on linearisation of vector-valued functions which means that vector-valued functions are represented by continuous linear operators. The first question we face is which vector-valued functions may be represented by continuous linear operators where vector-valued means that the functions have values in a locally convex Hausdorff space E. We study this problem in the framework of ε-products and give sufficient conditions when a space of E-valued functions coincides (up to an isomorphism) with the ε-product of a corresponding space of scalar-valued functions and the codomain E.
We apply our linearisation results to lift results that are known for the scalar-valued case to the vector-valued case. We transfer the solvability of a linear partial differential equation in certain function spaces from the scalar-valued case to the vector-valued case, which also gives an affirmative answer to the question of (continuous, smooth, holomorphic, distributional, etc.) parameter dependence of solutions in the scalar-valued case. Further, we give a unified approach to handle the problem of extending vector-valued functions via the existence of weak extensions under the constraint of preserving the properties, like holomorphy, of the scalar-valued extensions. Our approach also covers weak-strong principles. In particular, we study weak-strong principles for continuously partially differentiable functions of finite order and improve the well-known weak-strong principles of Grothendieck and Schwartz. We use our results to derive Blaschke's convergence theorem for several spaces of vector-valued functions and Wolff's theorem for the description of dual spaces of several function spaces of scalar-valued functions. Moreover, we transfer known series expansions and sequence space representations from scalar-valued to vector-valued functions.
We apply our linearisation results to lift results that are known for the scalar-valued case to the vector-valued case. We transfer the solvability of a linear partial differential equation in certain function spaces from the scalar-valued case to the vector-valued case, which also gives an affirmative answer to the question of (continuous, smooth, holomorphic, distributional, etc.) parameter dependence of solutions in the scalar-valued case. Further, we give a unified approach to handle the problem of extending vector-valued functions via the existence of weak extensions under the constraint of preserving the properties, like holomorphy, of the scalar-valued extensions. Our approach also covers weak-strong principles. In particular, we study weak-strong principles for continuously partially differentiable functions of finite order and improve the well-known weak-strong principles of Grothendieck and Schwartz. We use our results to derive Blaschke's convergence theorem for several spaces of vector-valued functions and Wolff's theorem for the description of dual spaces of several function spaces of scalar-valued functions. Moreover, we transfer known series expansions and sequence space representations from scalar-valued to vector-valued functions.
Subjects
vector-valued functions
weighted space
epsilon-product
linearisation
tensor product
extension
DDC Class
510: Mathematik
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