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Concept
Topologies on spaces of linear maps
Summary
In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves.
The article operator topologies discusses topologies on spaces of linear maps between normed spaces, whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs).
Throughout, the following is assumed:
is any non-empty set and is a non-empty collection of subsets of directed by subset inclusion (i.e. for any there exists some such that ).
is a topological vector space (not necessarily Hausdorff or locally convex).
is a basis of neighborhoods of 0 in
is a vector subspace of which denotes the set of all -valued functions with domain
The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
For any subsets and let
The family
forms a neighborhood basis
at the origin for a unique translation-invariant topology on where this topology is necessarily a vector topology (that is, it might not make into a TVS).
This topology does not depend on the neighborhood basis that was chosen and it is known as the topology of uniform convergence on the sets in or as the -topology.
However, this name is frequently changed according to the types of sets that make up (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details).
A subset of is said to be fundamental with respect to if each is a subset of some element in
In this case, the collection can be replaced by without changing the topology on
One may also replace with the collection of all subsets of all finite unions of elements of without changing the resulting -topology on
Call a subset of -bounded if is a bounded subset of for every
Properties
Properties of the basic open sets will now be described, so assume that and
Then is an absorbing subset of if and only if for all absorbs .
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Complete topological vector space
In 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.
Polar topology
In 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.
Mackey topology
In 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.