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Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.
Using this notion, a version of the spectral theorem for unbounded operators on Hilbert space can be formulated. [1] "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics." [2]
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.
The space is a fixed complex Hilbert space of countably infinite dimension. The observables of a quantum system are defined to be the (possibly unbounded ) self-adjoint operators A {\displaystyle A} on H {\displaystyle \mathbb {H} } .
Let and be Hilbert spaces, and let : be an unbounded operator from into . Suppose that is a closed operator and that is densely defined, that is, is dense in . Let : denote the adjoint of .
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
Normed space – Vector space on which a distance is defined; Operator algebra – Branch of functional analysis; Operator theory – Mathematical field of study; Topologies on the set of operators on a Hilbert space; Unbounded operator – Linear operator defined on a dense linear subspace
In the case where the Hilbert space is a space of functions on a bounded domain, these distinctions have to do with a familiar issue in quantum physics: One cannot define an operator—such as the momentum or Hamiltonian operator—on a bounded domain without specifying boundary conditions. In mathematical terms, choosing the boundary ...