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The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When V {\displaystyle V} is a complex vector space, it is assumed that for all v ≥ 0 , {\displaystyle v\geq 0,} f ( v ) {\displaystyle f(v)} is real.
Since positive linear functionals are bounded, the equivalence classes of the net {} converges to some vector in , which is a cyclic vector for . It is clear from the definition of the inner product on the GNS Hilbert space H {\displaystyle H} that the state ρ {\displaystyle \rho } can be recovered as a vector state on H {\displaystyle H} .
[2] [3] In functional analysis, the term linear functional is a synonym of linear form; [3] [4] [5] that is, it is a scalar-valued linear map. Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space . [citation needed]
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A proof can be sketched as follows: Let be the weak*-compact set of positive linear functionals on with norm ≤ 1, and () be the continuous functions on . A {\displaystyle A} can be viewed as a closed linear subspace of C ( Ω ) {\displaystyle C(\Omega )} (this is Kadison 's function representation ).
A weight ω on a von Neumann algebra is a linear map from the set of positive elements (those of the form a*a) to [0,∞]. A positive linear functional is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity). A state is a weight with ω(1) = 1. A trace is a weight with ω(aa*) = ω(a*a) for all a.
The Gelfand representation or Gelfand isomorphism for a commutative C*-algebra with unit is an isometric *-isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of A with the weak* topology.
There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or ...