In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space
into a preordered vector space
In other words, a positive linear operator maps the positive cone of the domain into the positive cone of the codomain.
Every positive linear functional is a type of positive linear operator.
The significance of positive linear operators lies in results such as Riesz–Markov–Kakutani representation theorem.
on a preordered vector space is called positive if it satisfies either of the following equivalent conditions: 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 preorder induced by the dual cone on the space of linear functionals on
is called the dual preorder.
be preordered vector spaces and let
be the space of all linear maps from
of all positive linear operators in
is a proper cone then this proper cone defines a canonical partial order on
into a partially ordered vector space.
are ordered topological vector spaces and if
is a family of bounded subsets of
, which is the space of all continuous linear maps from
it is sufficient that the positive cone of
(that is, the span of the positive cone of
is a locally convex space of dimension greater than 0 then this condition is also necessary.
is a locally convex space, then the canonical ordering of
are ordered locally convex topological vector spaces with
being a Mackey space on which every positive linear functional is continuous.
is a weakly normal cone in
is a barreled ordered topological vector space (TVS) with positive cone
is a semi-reflexive ordered TVS with a positive cone
that is directed upward and either majorized (that is, bounded above by some element of
exists and the section filter
uniformly on every precompact subset of