We introduce a new class FV(Ω, E) of weighted spaces of functions on a set Ω with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smooth, holomorphic or harmonic functions. Then we exploit the construction of FV(Ω, E) to derive sufficient conditions such that FV(Ω, E) can be linearised, i.e. that FV(Ω, E) is topologically isomorphic to the-product FV(Ω) E where FV(Ω) ∶= FV(Ω,) and is the scalar field of E.
We study spaces CV k (Ω, E) of k-times continuously partially differentiable functions on an open set Ω ⊂ R d with values in a locally convex Hausdorff space E. The space CV k (Ω, E) is given a weighted topology generated by a family of weights V k . For the space CV k (Ω, E) and its subspace CV k 0 (Ω, E) of functions that vanish at infinity in the weighted topology we try to answer the question whether their elements can be approximated by functions with values in a finite dimensional subspace. We derive sufficient conditions for an affirmative answer to this question using the theory of tensor products.
Let Ω be an open subset of R 2 and E a complete complex locally convex Hausdorff space. The purpose of this paper is to find conditions on certain weighted Fréchet spaces EV(Ω) of smooth functions and on the space E to ensure that the vector-valued Cauchy-Riemann operator ∂ : EV(Ω, E) → EV(Ω, E) is surjective. This is done via splitting theory and positive results can be interpreted as parameter dependence of solutions of the Cauchy-Riemann operator.
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