On vector-valued functions and the $\varepsilon$-product

Abstract

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 $\varepsilon$-products and give sufficient conditions when a space of $E$-valued functions coincides (up to an isomorphism) with the $\varepsilon$-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.