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Seppo I. Hiltunen

Publications and source records attributed to Seppo I. Hiltunen.

6 recordsLinked to original sources
arXiv2018

Duality of Bochner spaces

We construct the generalized Lebesgue--Bochner spaces $L^p(μ,\varPi)$ for positive measures $μ$ and for suitable real or complex topological vector spaces $\varPi$ so that for $1<p<+\infty$ and Banachable $\varPi$ with separable topology the strong dual of the classical Bochner space $L^p(μ,\varPi)$ becomes canonically represented by $L^{p^*}(μ,\varPi_σ')\,$. Hence we need no separability assumption of the norm topology of the strong dual $\varPi_β'$ of $\varPi$. For $p=1$ and for suitably restricted positive measures $μ$ we even get a similar result without any separability of the norm topology of the target space $\varPi$. For positive Radon measures on locally compact topological spaces these results are essentially contained on pages 588--606 in R. E. Edwards' classical Functional Analysis.

math.FA↗
arXiv2015

Real analyticity of composition is shy

Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $\mathbb S^{\,1.}\to\mathbb S^{\,1.}$ has a compatible analytic manifold structure, it does not make $G$ a real analytic Lie group since the group multiplication is not real analytic. The authors considered this result "surprising" for the applied concept of infinite-dimensional real analyticity for maps $E\to F$, defined by the property that locally a holomorphic extension $E_{\mathbb C}\to F_{\mathbb C}$ exist. In this note we show that this type of real analyticity is quite rare for composition maps ${\rm f\,}φ:x\mapstoφ\circ x$ when $φ$ is real analytic. Specifically, we show that the smooth Fréchet space map ${\rm f\,}φ:C\,(\mathbb R)\to C\,(\mathbb R)$ for real analytic $φ:\mathbb R\to\mathbb R$ is real analytic in the above sense only if $φ$ is the restriction to $\mathbb R$ of some entire function $\mathbb C\to\mathbb C$. We also discuss the possibility of proving that the set of these "admissible" functions $φ$ be "small" in the space $A\,(\mathbb R)$ of real analytic functions either in the Baire categorical sense, or in the measure theoretic sense of shyness.

math.FA↗
arXiv2007

Seip's differentiability concepts as a particular case of the Bertram--Gloeckner--Neeb construction

From the point of view of unification of differentiation theory, it is of interest to note that the general construction principle of Bertram, Gloeckner and Neeb leading to a C^k differentiability concept from a given C^0 one, besides subsuming the Keller--Bastiani C_c^k differentiabilities on real Hausdorff locally convex spaces, also does the same to the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl, and likewise to the "compactly generated" theory of Seip's continuous differentiabilities. In this article, we give the details of the proof for the assertion concerning Seip's theory. We also give an example indicating that the premises in Seip's various inverse and implicit function theorems may be too strong in order for these theorems to have much practical value. Also included is a presentation of the BGN--setting reformulated so as to be consistent with the Kelley--Morse--Godel--Bernays--von Neumann type approach to set theory, as well as a treatment of the function space constructions and development of their basic properties needed in the proof of the main result.

math.FA↗
arXiv2007

On Yamamuro's inverse and implicit function theorems in terms of calibrations

For the Frechet space E=C^{\infty}(S^1) and for a smooth ϕ: R to R, we prove that the associated map E to E given by x mapstoϕ\circ x satisfies the continuous BΓ--differentiability condition in Yamamuro's inverse function theorem only if ϕis affine. Via more complicated examples, we also generally discuss the importance of testing the applicability of proposed inverse and implicit function theorems by this kind of simple maps.

math.FA↗
arXiv2007

An inverse function theorem for Colombeau tame Frolicher-Kriegl maps

For k=1,2,... infty and a Frolicher-Kriegl order k Lipschitz differentiable map f:E supseteq U to E having derivative at x_0 in U a linear homeomorphism E to E and satisfying a Colombeau type tameness condition, we prove that x_0 has a neighborhood V subseteq U with f|V a local order k Lipschitz diffeomorphism. As a corollary we obtain a similar result for Keller C_c^{\infty} maps with E in a class including Frechet and Silva spaces. We also indicate a procedure for verifying the tameness condition for maps of the type x mapsto varphi circ [id,x] and spaces E=C^{\infty}(Q) when Q is compact by considering the case Q=[0,1]. Our considerations are motivated by the wish to try to retain something valuable in an interesting but defective treatment of integrability of Lie algebras by J. Leslie.

math.FA↗