In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X {\displaystyle X} is the continuous dual space X ′ {\displaystyle X^{\prime }} of X {\displaystyle X} equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of X , {\displaystyle X,} where this topology is denoted by b ( X ′ , X ) {\displaystyle b\left(X^{\prime },X\right)} or β ( X ′ , X ) . {\displaystyle \beta \left(X^{\prime },X\right).} 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, X ′ , {\displaystyle X^{\prime },} has the strong dual topology, X b ′ {\displaystyle X_{b}^{\prime }} or X β ′ {\displaystyle X_{\beta }^{\prime }} may be written.
Strong dual topology Throughout, all vector spaces will be assumed to be over the field F {\displaystyle \mathbb {F} } of either the real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .}
Definition from a dual system
Let ( X , Y , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (X,Y,\langle \cdot ,\cdot \rangle )} be a dual pair of vector spaces over the field F {\displaystyle \mathbb {F} } of real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .} For any B ⊆ X {\displaystyle B\subseteq X} and any y ∈ Y , {\displaystyle y\in Y,} define
| y | B = sup x ∈ B | ⟨ x , y ⟩ | . {\displaystyle |y|_{B}=\sup _{x\in B}|\langle x,y\rangle |.}
Neither X {\displaystyle X} nor Y {\displaystyle Y} has a topology so say a subset B ⊆ X {\displaystyle B\subseteq X} is said to be bounded by a subset C ⊆ Y {\displaystyle C\subseteq Y} if | y | B < ∞ {\displaystyle |y|_{B}<\infty } for all y ∈ C . {\displaystyle y\in C.} So a subset B ⊆ X {\displaystyle B\subseteq X} is called bounded if and only if
sup x ∈ B | ⟨ x , y ⟩ | < ∞ for all y ∈ Y . {\displaystyle \sup _{x\in B}|\langle x,y\rangle |<\infty \quad {\text{ for all }}y\in Y.} This is equivalent to the usual notion of bounded subsets when X {\displaystyle X} is given the weak topology induced by Y , {\displaystyle Y,} which is a Hausdorff locally convex topology. Let B {\displaystyle {\mathcal {B}}} denote the family of all subsets B ⊆ X {\displaystyle B\subseteq X} bounded by elements of Y {\displaystyle Y} ; that is, B {\displaystyle {\mathcal {B}}} is the set of all subsets B ⊆ X {\displaystyle B\subseteq X} such that for every y ∈ Y , {\displaystyle y\in Y,}
| y | B = sup x ∈ B | ⟨ x , y ⟩ | < ∞ . {\displaystyle |y|_{B}=\sup _{x\in B}|\langle x,y\rangle |<\infty .}
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