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2017 | OriginalPaper | Chapter

On Repeated Sequential Closures of Constructible Functions in Valuations

Author : Semyon Alesker

Published in: Geometric Aspects of Functional Analysis

Publisher: Springer International Publishing

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Abstract

The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in Alesker (Geom Funct Anal 20(5):1073–1143, 2010).

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next chapter Orbit Point of View on Some Results of Asymptotic Theory; Orbit Type and Cotype
Footnotes
1
This fact was pointed out to the author by C. Brouder in September 2013. I am very grateful to him for this remark.
 
2
This theorem says that if \(K,L \in \mathcal{K}(V )\) cannot be separated by a hyperplane (i.e. there is no hyperplane such that K and L are contained in different closed subspaces defined by the hyperplane) and if convex compact sets K i  → K and L i  → L in the Hausdorff metric as i → ∞, then K i ∩ L i  → K ∩ L.
 
3
This canonical isomorphism was proved in Lemma 5.1.3(1) of [5]. \(C_{Z_{1}}^{\infty }(V,V al_{i}^{\infty }(V ))\) denotes the space of smooth functions on V with support in Z1 and with valued in the Fréchet space V ali ∞ (V ).
 
Literature
1.
S. Alesker, Integrals of smooth and analytic functions over Minkowski’s sums of convex sets. Convex geometric analysis (Berkeley, CA, 1996), pp, 1–15. Math. Sci. Res. Inst. Publ, vol. 34 (Cambridge University Press, Cambridge, 1999)
2.
S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11 (2), 244–272 (2001)MathSciNetCrossRefMATH
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5.
S. Alesker, Theory of valuations on manifolds. IV. New properties of the multiplicative structure, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1910 (Springer, Berlin, 2007), pp. 1–44
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S. Alesker, A. Bernig, The product on smooth and generalized valuations. Am. J. Math. 134 (2), 507–560 (2012). Also:arXiv:0904.1347
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C. Brouder, N.V. Dang, F. Hélein, Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker). Studia Math. 232 (3), 201–226 (2016)MathSciNetMATH
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M. Kashiwara, P. Schapira, Sheaves on manifolds. With a chapter in French by Christian Houzel. Corrected reprint of the 1990 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292 (Springer, Berlin, 1994)
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A.G. Khovanski, A.V. Pukhlikov, Finitely additive measures of virtual polyhedra. (Russian) Algebra i Analiz 4 (2), 161–185 (1992). Translation in St. Petersburg Math. J. 4 (2), 337–356 (1993)
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V.A. Zorich, Mathematical Analysis. II. Universitext (Springer, Berlin, 2004). Translated from the 2002 fourth Russian edition by Roger Cooke
Metadata
Title
On Repeated Sequential Closures of Constructible Functions in Valuations
Author
Semyon Alesker
Copyright Year
2017
Book
Geometric Aspects of Functional Analysis

Print ISBN: 978-3-319-45281-4

Electronic ISBN: 978-3-319-45282-1

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-45282-1_1

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