# 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.

## 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 ${\displaystyle p\wedge (e-p)=0}$. If ${\displaystyle p_,p_,\ldots ,p_}$ are pairwise disjoint components of e, any real linear combination of ${\displaystyle p_,p_,\ldots ,p_}$ 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 ${\displaystyle \\}}$ and ${\displaystyle \\}}$ of e-simple functions, such that ${\displaystyle \\}}$ is monotone increasing and converges e-uniformly to f, and ${\displaystyle \\}}$ is monotone decreasing and converges e-uniformly to f.