Freudenthal spectral theorem
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Freudenthal spectral theorem is a result in Riesz space theory proved by
Hans Freudenthal Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education ...
in 1936. It roughly states that any element dominated by a positive element in a
Riesz space In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Su ...
with the principal projection property can in a sense be approximated uniformly by
simple function In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, ...
s. Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known
Radon–Nikodym theorem In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
, the validity of the Poisson formula and the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
from the theory of
normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operat ...
s can all be shown to follow as special cases of the Freudenthal spectral theorem.


Statement

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 \ and \ of ''e''-simple functions, such that \ is monotone increasing and converges ''e''-uniformly to ''f'', and \ is monotone decreasing and converges ''e''-uniformly to ''f''.


Relation to the Radon–Nikodym theorem

Let (X,\Sigma) be a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
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 In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ' ...
, 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 function In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, ...
s 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 In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Inform ...
\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

*
Radon–Nikodym theorem In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...


References

* * {{Ordered topological vector spaces Theorems in functional analysis