Polar setIn functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in (not ). There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a pairing of vector spaces over the real or complex numbers ( and are often topological vector spaces (TVSs)).
Mackey topologyIn functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. A topological vector space (TVS) is called a Mackey space if its topology is the same as the Mackey topology.
Continuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.
Convex coneIn linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, C is a cone if implies for every positive scalar s. When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a positive scalar.
Strong dual spaceIn functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space of equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of where this topology is denoted by or The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise.
Polar topologyIn functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.
Initial topologyIn general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set with respect to a family of functions on is the coarsest topology on that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.
Convex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex set A subset of some vector space is if it satisfies any of the following equivalent conditions: If is real and then If is real and with then Convex function Throughout, will be a map valued in the extended real numbers with a domain that is a convex subset of some vector space.
Complete topological vector spaceIn functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point towards which they all get closer" means that this Cauchy net or filter converges to The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces.
EquicontinuityIn mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous.
