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
In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.
When is a complex vector space, it is assumed that for all is real. As in the case when is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace and the partial order does not extend to all of in which case the positive elements of are the positive elements of by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any equal to for some to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.
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specifically in functional analysis, a positivelinearfunctional on an ordered vector space ( V , ≤ ) {\displaystyle (V,\leq )} is a linearfunctional f {\displaystyle...
In mathematics, a linear form (also known as a linearfunctional, a one-form, or a covector) is a linear map from a vector space to its field of scalars...
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation...
In mathematics, more specifically in functional analysis, a positivelinear operator from an preordered vector space ( X , ≤ ) {\displaystyle (X,\leq )}...
implies a = 0. Every element Ω of H defines a positivelinearfunctional ωΩ on a *-algebra A of bounded linear operators in H by the relation ωΩ(a) = (aΩ...
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X → Y {\displaystyle L:X\to Y} between topological...
such a linear function from the other concept, the term affine function is often used. In linear algebra, mathematical analysis, and functional analysis...
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional...
(controls), a term related to control theory State (functional analysis), a positivelinearfunctional on an operator algebra State, in dynamical systems...
Haar measure as a by-product. The functional μ A {\displaystyle \mu _{A}} extends to a positivelinearfunctional on compactly supported continuous functions...
be defined, by the Riesz representation theorem, by giving a positivelinearfunctional Λ on the space C0(M) of compactly supported continuous functions...
(primitive causality). A state with respect to a C*-algebra is a positivelinearfunctional over it with unit norm. If we have a state over A ( M ) {\displaystyle...
equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets...
Hausdorff spaces, and only consider the measures that correspond to positivelinearfunctionals on the space of continuous functions with compact support (some...
Neumann algebra is a linear map from the set of positive elements (those of the form a*a) to [0,∞]. A positivelinearfunctional is a weight with ω(1)...
The generalized functionallinear model (GFLM) is an extension of the generalized linear model (GLM) that allows one to regress univariate responses of...
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is...
In mathematics, particularly in functional analysis, a seminorm is a norm that need not be positive definite. Seminorms are intimately connected with...
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