Vanish at infinityIn mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces. Aside from this difference, both of these notions correspond to the intuitive notion of adding a point at infinity, and requiring the values of the function to get arbitrarily close to zero as one approaches it.
Injective tensor productIn mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the . Injective tensor products have applications outside of nuclear spaces.
Metrizable topological vector spaceIn functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
Distinguished spaceIn functional analysis and related areas of mathematics, 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 strong dual space) are contained in the weak-* closure of some bounded subset of the bidual. Suppose that is a locally convex space and let and denote the strong dual of (that is, the continuous dual space of endowed with the strong dual topology).
Complete topological vector spaceIn functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point towards which they all get closer" means that this Cauchy net or filter converges to The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces.
Mackey spaceIn mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X′), the finest topology which still preserves the continuous dual. They are named after George Mackey. Examples of locally convex spaces that are Mackey spaces include: All barrelled spaces and more generally all infrabarreled spaces Hence in particular all bornological spaces and reflexive spaces All metrizable spaces.
Espace nucléaireEn mathématiques, et plus précisément en analyse, un espace nucléaire est un espace vectoriel topologique possédant certaines propriétés analogues à celles des espaces de dimension finie. Leur topologie peut être définie par une famille de semi-normes dont la taille des boules unités décroit rapidement. Les espaces vectoriels dont les éléments sont « lisses » en un certain sens sont souvent des espaces nucléaires ; un exemple typique est celui des fonctions régulières sur une variété compacte.
Théorème de représentation de Riesz (Fréchet-Riesz)En mathématiques, plus précisément en analyse fonctionnelle, le théorème de représentation de Riesz, en l'honneur du mathématicien Frigyes Riesz, est un théorème qui représente les éléments du dual d'un espace de Hilbert comme produit scalaire par un vecteur de l'espace. Ce théorème est aussi parfois appelé théorème de Fréchet-Riesz (à ne pas confondre avec le théorème de Riesz-Fréchet-Kolmogorov). Il s'apparente singulièrement au théorème de Lax-Milgram qui englobe l'énoncé ci-dessous.
Espace de Schwartzvignette|Une fonction gaussienne bidimensionnelle est un exemple de fonction à décroissance rapide. En analyse mathématique, l'espace de Schwartz est l'espace des fonctions déclinantes (c'est-à-dire des fonctions indéfiniment dérivables à décroissance rapide, ainsi que leurs dérivées de tous ordres). Le dual de cet espace est l'espace des distributions tempérées. Les espaces et jouent un rôle essentiel dans la théorie de la transformée de Fourier.
Continuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.
