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In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (,) is a linear functional on so that for all positive elements, that is , it holds that ()
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} .
A linear function on a preordered vector space is called positive if it satisfies either of the following equivalent conditions: . implies (); if then () (). [1]; The set of all positive linear forms on a vector space with positive cone , called the dual cone and denoted by , is a cone equal to the polar of .
The statement of the theorem for positive linear functionals on C c (X), the space of compactly supported complex-valued continuous functions, is as follows: . Theorem Let X be a locally compact Hausdorff space and a positive linear functional on C c (X).
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states.
Let be a real vector space, be a vector subspace, and be a convex cone.. A linear functional: is called -positive, if it takes only non-negative values on the cone : ().A linear functional : is called a -positive extension of , if it is identical to in the domain of , and also returns a value of at least 0 for all points in the cone :
(The positive and negative parts of a self-adjoint operator are obtained by the continuous functional calculus.) The trace is a linear functional over the space of trace-class operators, that is, (+) = + ().
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear function on is called positive if for all , implies () [1] The order dual of is denoted by +.