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 ${\displaystyle p\wedge (e-p)=0}$. If ${\displaystyle p_{1},p_{2},\ldots ,p_{n}}$ are pairwise disjoint components of e, any real linear combination of ${\displaystyle p_{1},p_{2},\ldots ,p_{n}}$ is called an e-simple function.

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