Strong topology (polar topology)
Encyclopedia
In functional analysis
and related areas of mathematics
the strong topology is the finest polar topology, the topology
with the most open set
s, on a dual pair
. The coarsest polar topology is called weak topology
.
the strong topology 
on 
is the polar topology defined by using the family of all sets in 
where the polar set in 
is absorbent
.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
the strong topology is the finest polar topology, the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
with the most open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s, on a dual pair
Dual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
. The coarsest polar topology is called weak topology
Weak topology (polar topology)
In functional analysis and related areas of mathematics the weak topology is the coarsest polar topology, the topology with the fewest open sets, on a dual pair...
.
Definition
Given a dual pairDual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
the strong topology on is the polar topology defined by using the family of all sets in where the polar set in is absorbentAbsorption law
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.Two binary operations, say ¤ and *, are said to be connected by the absorption law if:...
.
Examples
- Given a normed vector spaceNormed vector spaceIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
and its continuous dual 
then 
-topology on 
is identical to the topology induced by the operator normOperator normIn mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
. Conversely
-topology on 
is identical to the topology induced by the normNorm (mathematics)In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
.
Properties
- In barrelled spaceBarrelled spaceIn functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced,...
s the strong topology is identical to the Mackey topologyMackey 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...
.