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...

distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their...

(which is the strong dual of the strong dual of X {\displaystyle X} ) is a homeomorphism (or equivalently, a TVS isomorphism). A normed space is reflexive...

In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms...

semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of...

metrizable strong dual spaces. Every normed space can be isometrically embedded onto a dense vector subspace of a Banach space, where this Banach space is called...

the origin. the strong dual space X b ′ {\displaystyle X_{b}^{\prime }} of X {\displaystyle X} is normable. the strong dual space X b ′ {\displaystyle X_{b}^{\prime...

Fréchet space. The strong dual of a reflexive Fréchet space is a bornological space and a Ptak space. Every Fréchet space is a Ptak space. The strong bidual...

instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and...

is the continuous dual space of X {\displaystyle X} endowed with the strong dual topology). A locally convex topological vector space (TVS) X {\displaystyle...

𝒮(Rn) and its strong dual space are also: complete Hausdorff locally convex spaces, nuclear Montel spaces, ultrabornological spaces, reflexive barrelled...

a DF-space. The strong dual of a DF-space is a Fréchet space. The strong dual of a reflexive Fréchet space is a bornological space. The strong bidual...

C^{\infty }(U),} as well as the strong dual spaces of both these of spaces, are complete nuclear Montel ultrabornological spaces, which implies that all four...

space C c ∞ {\displaystyle C_{c}^{\infty }} with L 2 {\displaystyle L^{2}} (which is a reflexive space that is even isomorphic to its own strong dual...

topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets Reductive dual pair Strong dual space – Continuous dual space endowed...

Hausdorff locally convex TVS. The strong dual space of C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} is called the space of distributions on U {\displaystyle...

{D}}'(U)} are Montel spaces and, in the dual space of any Montel space, a sequence of continuous linear functionals converges in the strong dual topology if and...

distinguished spaces, DF-spaces, and σ {\displaystyle \sigma } -barrelled spaces that are not quasibarrelled. The strong dual space X b ′ {\displaystyle X_{b}^{\prime...

\ell ^{p}} space with the largest p {\displaystyle p} . This space is the strong dual space of ℓ 1 {\displaystyle \ell ^{1}} : indeed, every x ∈ ℓ ∞ {\displaystyle...

Fréchet space if and only if all Xi are normable. Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space. A typical...

LF-spaces such as the space of test functions C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} with it canonical LF-topology, the strong dual space of any...

nuclear Montel bornological barrelled Mackey space; the same is true of its strong dual space (that is, the space of all distributions with its usual topology)...

bornological strong duals. The strong dual of every reflexive Fréchet space is bornological. If the strong dual of a metrizable locally convex space is separable...

values of the dual and primal LPs. The strong duality theorem states that, moreover, if the primal has an optimal solution then the dual has an optimal...

initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal...

quasi-complete Schwartz space is a semi-Montel space. Every Fréchet Schwartz space is a Montel space. The strong dual space of a complete Schwartz space is an ultrabornological...

vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again...

the Gelfand dual of A (not to be confused with the dual A' of the Banach space A). In particular, suppose X is a compact Hausdorff space. Then there is...

else that they are both DF-spaces. Then, denoting strong dual spaces with a subscripted b {\displaystyle b} : The strong dual of N ⊗ ^ π Y {\displaystyle...

Weak convergence (Hilbert space) Weak* topology Polar topology Strong dual space Strong operator topology Topologies on spaces of linear maps Ultrastrong...