Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. It's essentially just an infinitely dimensional vector space. ﻿A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
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