World Library  
Flag as Inappropriate
Email this Article

Freudenthal spectral theorem

Article Id: WHEBN0034121965
Reproduction Date:

Title: Freudenthal spectral theorem  
Author: World Heritage Encyclopedia
Language: English
Subject: Functional analysis, Symmetric set, Radial set, Strictly singular operator, Ultrastrong topology
Publisher: World Heritage Encyclopedia

Freudenthal spectral theorem

In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sense be approximated uniformly by simple functions.

Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known Radon–Nikodym theorem, the validity of the Poisson formula and the spectral theorem from the theory of normal operators can all be shown to follow as special cases of the Freudenthal spectral theorem.


Let e be any positive element in a Riesz space E. A positive element of p in E is called a component of e if p\wedge(e-p)=0. If p_1,p_2,\ldots,p_n are pairwise disjoint components of e, any real linear combination of p_1,p_2,\ldots,p_n is called an e-simple function.

The Freudenthal spectral theorem states: Let E be any Riesz space with the principal projection property and e any positive element in E. Then for any element f in the principal ideal generated by e, there exist sequences \{s_n\} and \{t_n\} of e-simple functions, such that \{s_n\} is monotone increasing and converges e-uniformly to f, and \{t_n\} is monotone decreasing and converges e-uniformly to f.

Relation to the Radon–Nikodym theorem

Let (X,\Sigma) be a measure space and M_\sigma the real space of signed \sigma-additive measures on (X,\Sigma). It can be shown that M_\sigma is a Dedekind complete Banach Lattice with the total variation norm, and hence has the principal projection property. For any positive measure \mu, \mu-simple functions (as defined above) can be shown to correspond exactly to \mu-measurable simple functions on (X,\Sigma) (in the usual sense). Moreover, since by the Freudenthal spectral theorem, any measure \nu in the band generated by \mu can be monotonously approximated from below by \mu-measurable simple functions on (X,\Sigma), by Lebesgue's monotone convergence theorem \nu can be shown to correspond to an L^1(X,\Sigma,\mu) function and establishes an isometric lattice isomorphism between the band generated by \mu and the Banach Lattice L^1(X,\Sigma,\mu).

See also


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.