Functional analysis
Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to formula_1. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven.
Observable
If the Hilbert space is finite-dimension, it can be represented by a Hermitian matrix if the space is finite-dimensional. In an infinite-dimensional Hilbert space, the observable is represented by a symmetric operator, which may not be defined everywhere. The reason for such a change is that in an infinite-dimensional Hilbert space, the observable operator can become unbounded, which means that it no longer has a largest eigenvalue. This is not the case in a finite-dimensional Hilbert space: an operator can have no more eigenvalues than the dimension of the state it acts upon, and by the well-ordering property, any finite set of real numbers has a largest element. For example, the position of a point particle moving along a line can take any real number as its value, and the set of real numbers is uncountably infinite. Since the eigenvalue of an observable represents a possible physical quantity that its corresponding dynamical variable can take, we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space. 
A crucial difference between classical quantities and quantum mechanical observables is that the latter may not be simultaneously measurable, a property referred to as complementarity. This is mathematically expressed by non-commutativity of the corresponding operators, to the effect that the commutator