Throughout, all vector spaces will be assumed to be over the field \mathbb{F} of either the
real numbers \R or
complex numbers \C.
Definition from a dual system Let (X, Y, \langle \cdot, \cdot \rangle) be a
dual pair of vector spaces over the field \mathbb{F} of
real numbers \R or
complex numbers \C. For any B \subseteq X and any y \in Y, define |y|_B = \sup_{x \in B}|\langle x, y\rangle|. Neither X nor Y has a topology so say a subset B \subseteq X is said to be '''''' if |y|_B for all y \in C. So a subset B \subseteq X is called if and only if \sup_{x \in B} |\langle x, y \rangle| This is equivalent to the usual notion of
bounded subsets when X is given the weak topology induced by Y, which is a Hausdorff
locally convex topology. Let \mathcal{B} denote the
family of all subsets B \subseteq X bounded by elements of Y; that is, \mathcal{B} is the set of all subsets B \subseteq X such that for every y \in Y, |y|_B = \sup_{x\in B}|\langle x, y\rangle| Then the '''''' \beta(Y, X, \langle \cdot, \cdot \rangle) on Y, also denoted by b(Y, X, \langle \cdot, \cdot \rangle) or simply \beta(Y, X) or b(Y, X) if the pairing \langle \cdot, \cdot\rangle is understood, is defined as the
locally convex topology on Y generated by the seminorms of the form |y|_B = \sup_{x \in B} |\langle x, y\rangle|,\qquad y \in Y, \qquad B \in \mathcal{B}. The definition of the strong dual topology now proceeds as in the case of a TVS. Note that if X is a TVS whose continuous dual space
separates points on X, then X is part of a canonical dual system \left(X, X^{\prime}, \langle \cdot , \cdot \rangle\right) where \left\langle x, x^{\prime} \right\rangle := x^{\prime}(x). In the special case when X is a
locally convex space, the '''''' on the (continuous)
dual space X^{\prime} (that is, on the space of all continuous linear functionals f : X \to \mathbb{F}) is defined as the strong topology \beta\left(X^{\prime}, X\right), and it coincides with the topology of uniform convergence on
bounded sets in X, i.e. with the topology on X^{\prime} generated by the seminorms of the form |f|_B = \sup_{x \in B} |f(x)|, \qquad \text{ where } f \in X^{\prime}, where B runs over the family of all
bounded sets in X. The space X^{\prime} with this topology is called '''''' of the space X and is denoted by X^{\prime}_{\beta}.
Definition on a TVS Suppose that X is a
topological vector space (TVS) over the field \mathbb{F}. Let \mathcal{B} be any fundamental system of
bounded sets of X; that is, \mathcal{B} is a
family of bounded subsets of X such that every bounded subset of X is a subset of some B \in \mathcal{B}; the set of all bounded subsets of X forms a fundamental system of bounded sets of X. A basis of closed neighborhoods of the origin in X^{\prime} is given by the
polars: B^{\circ} := \left\{ x^{\prime} \in X^{\prime} : \sup_{x \in B} \left|x^{\prime}(x)\right| \leq 1 \right\} as B ranges over \mathcal{B}). This is a locally convex topology that is given by the set of
seminorms on X^{\prime}: \left|x^{\prime}\right|_{B} := \sup_{x \in B} \left|x^{\prime}(x)\right| as B ranges over \mathcal{B}. If X is
normable then so is X^{\prime}_{b} and X^{\prime}_{b} will in fact be a
Banach space. If X is a normed space with norm \| \cdot \| then X^{\prime} has a canonical norm (the
operator norm) given by \left\| x^{\prime} \right\| := \sup_{\| x \| \leq 1} \left| x^{\prime}(x) \right|; the topology that this norm induces on X^{\prime} is identical to the strong dual topology. == Bidual ==