Throughout, the following is assumed: The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
forms a neighborhood basis[1] at the origin for a unique translation-invariant topology on
that was chosen and it is known as the topology of uniform convergence on the sets in
[2] However, this name is frequently changed according to the types of sets that make up
with the collection of all subsets of all finite unions of elements of
forms a fundamental system of entourages for a uniform structure on
is a family of continuous seminorms generating this topology on
is a non-trivial completely regular Hausdorff topological space and
is the space of all real (or complex) valued continuous functions on
will denote the vector space of all continuous linear maps from
(which we will assume to be real or complex numbers) is the vector space
Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.
The following assumption is very commonly made because it will guarantee that each set
are locally convex, then we may add to this list: Common assumptions Some authors (e.g. Narici) require that
is directed by subset inclusion: Some authors (e.g. Trèves [9]) require that
be directed under subset inclusion and that it satisfy the following condition: If
consisting of all continuous linear maps that are bounded on every
is directed by subset inclusion, and satisfies the following condition: if
are locally convex Hausdorff spaces then By letting
or the topology of uniform convergence on bounded sets and
Consequently, in this case the results mentioned in this article can be applied to polar topologies.
(as defined in this article) is not necessarily a polar topology.
One important difference is that polar topologies are always locally convex while
denote the space of separately continuous bilinear maps and
denote the space of continuous bilinear maps, where
are topological vector space over the same field (either the real or complex numbers).
However, as before, this topology is not necessarily compatible with the vector space structure of
consist of bounded sets then this requirement is automatically satisfied if we are topologizing
consists of bounded sets and any of the following conditions hold: Suppose that
Part of the importance of this vector space and this topology is that it contains many subspace, such as