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In 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.
To emphasize that the continuous dual space, has the strong dual topology, or may be written.

Throughout, all vector spaces will be assumed to be over the field of either the real numbers or complex numbers

Let be a dual pair of vector spaces over the field of real numbers or complex numbers For any and any define

Neither nor has a topology so say a subset is said to be ** bounded by a subset ** if for all
So a subset is called

Let denote the family of all subsets bounded by elements of ; that is, is the set of all subsets such that for every

The definition of the strong dual topology now proceeds as in the case of a TVS.
Note that if is a TVS whose continuous dual space separates point on then is part of a canonical dual system
where
In the special case when is a locally convex space, the ** strong topology** on the (continuous) dual space (that is, on the space of all continuous linear functionals ) is defined as the strong topology and it coincides with the topology of uniform convergence on bounded sets in i.e. with the topology on generated by the seminorms of the form

Suppose that is a topological vector space (TVS) over the field Let be any fundamental system of bounded sets of ; that is, is a family of bounded subsets of such that every bounded subset of is a subset of some ; the set of all bounded subsets of forms a fundamental system of bounded sets of A basis of closed neighborhoods of the origin in is given by the polars:

If is normable then so is and will in fact be a Banach space. If is a normed space with norm then has a canonical norm (the operator norm) given by ; the topology that this norm induces on is identical to the strong dual topology.

The **bidual** or **second dual** of a TVS often denoted by is the strong dual of the strong dual of :

Let be a locally convex TVS.

- A convex balanced weakly compact subset of is bounded in
^{[1]} - Every weakly bounded subset of is strongly bounded.
^{[2]} - If is a barreled space then 's topology is identical to the strong dual topology and to the Mackey topology on
- If is a metrizable locally convex space, then the strong dual of is a bornological space if and only if it is an infrabarreled space, if and only if it is a barreled space.
^{[3]} - If is Hausdorff locally convex TVS then is metrizable if and only if there exists a countable set of bounded subsets of such that every bounded subset of is contained in some element of
^{[4]} - If is locally convex, then this topology is finer than all other -topologies on when considering only 's whose sets are subsets of
- If is a bornological space (e.g. metrizable or LF-space) then is complete.

If is a barrelled space, then its topology coincides with the strong topology on and with the Mackey topology on generated by the pairing

If is a normed vector space, then its (continuous) dual space with the strong topology coincides with the Banach dual space ; that is, with the space with the topology induced by the operator norm. Conversely -topology on is identical to the topology induced by the norm on

- Dual topology
- Dual system
- List of topologies – List of concrete topologies and topological spaces
- Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
- Reflexive space – Locally convex topological vector space
- Semi-reflexive space
- Strong topology
- Topologies on spaces of linear maps

**^**Schaefer & Wolff 1999, p. 141.**^**Schaefer & Wolff 1999, p. 142.**^**Schaefer & Wolff 1999, p. 153.**^**Narici & Beckenstein 2011, pp. 225–273.

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