Solèr's theorem


In mathematics, Solèr's theorem is a result concerning certain infinite-dimensional vector spaces. It states that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions. Originally proved by Maria Pia Solèr, the result is significant for quantum logic and the foundations of quantum mechanics. In particular, Solèr's theorem helps to fill a gap in the effort to use Gleason's theorem to rederive quantum mechanics from information-theoretic postulates.
Physicist John C. Baez notes,
Nothing in the assumptions mentions the continuum: the hypotheses are purely algebraic. It therefore seems quite magical that is forced to be the real numbers, complex numbers or quaternions.
Writing a decade after Solèr's original publication, Pitowsky calls her theorem "celebrated".

Statement

Let be a division ring. That means it is a ring in which one can add, subtract, multiply, and divide but in which the multiplication need not be commutative. Suppose this ring has a conjugation, i.e. an operation for which
Consider a vector space V with scalars in, and a mapping
which is -linear in left entry, satisfying the identity
This is called a Hermitian form. Suppose this form is non-degenerate in the sense that
For any subspace S let be the orthogonal complement of S. Call the subspace "closed" if
Call this whole vector space, and the Hermitian form, "orthomodular" if for every closed subspace S we have that is the entire space.
A set of vectors is called "orthonormal" if The result is this: