Subsequent authors have shown that this version of the theorem is not generally true, however.

An easier to state version of the theorem uses graph theory.

A precise version of the spectral theorem in this case is:

This work at the end of the 19th century opened into several successive versions of the theorem.

Other versions of the theorem involving the weak or strong energy condition also exist.

In the first version of the theorem, evidently the separating hyperplane is never unique.

Still, a modified version of the general theorem can be proved.

This position is a more refined version of the theorem first discovered by David Hume.

A version of the above theorem holds locally in the following sense .

A more sophisticated version of the theorem involving fields states that: