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 bounded if and only if This is equivalent to the usual notion of bounded subsets when is given the weak topology induced by which is a Hausdorff locally convex topology.

Let denote the family of all subsets bounded by elements of ; that is, is the set of all subsets such that for every Then the strong topology on also denoted by or simply or if the pairing is understood, is defined as the locally convex topology on generated by the seminorms of the form

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 where runs over the family of all bounded sets in The space with this topology is called strong dual space of the space and is denoted by

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: as ranges over ). This is a locally convex topology that is given by the set of seminorms on : as ranges over

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 : where denotes endowed with the strong dual topology Unless indicated otherwise, the vector space is usually assumed to be endowed with the strong dual topology induced on it by in which case it is called the strong bidual of ; that is, where the vector space is endowed with the strong dual topology

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

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