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In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.

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rdf:type	Thing
rdfs:label	Halbnorm (de) Seminorma (es) Semi-norme (fr) Seminorma (it) 半ノルム (ja) Seminorm (en) Seminorma (pt) Полунорма (ru) Напівнорма (uk)
rdfs:comment	In der Mathematik versteht man unter einer Halbnorm (oder unter einer Seminorm) ein Funktional, das sowohl absolut homogen als auch subadditiv ist. Mit dem Konzept der Halbnorm wird das Konzept der Norm verallgemeinert, indem auf die Eigenschaft der positiven Definitheit verzichtet wird. Jede Halbnorm ist nichtnegativ, symmetrisch bezüglich Vorzeichenumkehr, sublinear und konvex. Aus jeder Halbnorm kann durch Restklassenbildung eine zugehörige Norm abgeleitet werden. Mit Hilfe von Familien von Halbnormen können auch lokalkonvexe Vektorräume definiert werden. Halbnormen werden insbesondere in der linearen Algebra und in der Funktionalanalysis studiert. Eng verwandt mit dem Konzept der Halbnorm ist das Konzept des Minkowski-Funktionals. (de) Una seminorma es una generalización del concepto de norma vectorial que se define a partir de un producto escalar el cual no es definido positivo o se dice que es semiescalar. (es) In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms. (en) In algebra lineare, una seminorma è una generalizzazione del concetto di norma che, a differenza di quest'ultima, può assegnare lunghezza zero anche ad un vettore diverso da zero. La nozione di seminorma è utilizzata in vari ambiti dell'analisi funzionale. Una famiglia numerabile di seminorme, per esempio, consente di indurre una topologia su uno spazio di Fréchet. (it) 数学の特に線型代数学および函数解析学における半ノルム（はんのるむ、英: seminorm, semi-norm; セミノルム）は、ベクトル空間上で定義される絶対斉次劣加法的函数で、正定値と制約しないことによるノルムの一般化である。半ノルムの値は非負かつ符号反転に関してであり、函数としてかつ凸である。各半ノルムには、適当な剰余類をとる商構成に誘導されるノルムが付随する。半ノルムからなる族を用いて、局所凸線型空間を定義することができる。 (ja) Em matemática, uma seminorma consiste numa função que associa cada vetor de um espaço vetorial em um número real não negativo. (pt) Напівнорма або переднорма — узагальнення поняття норми; на відміну від норми напівнорма може бути рівною нулю на ненульових елементах простору. (uk) Полунорма или преднорма — обобщение понятия норма; в отличие от последней, полунорма может равняться нулю на ненулевых элементах пространства. (ru) En mathématiques, une semi-norme est une application d'un espace vectoriel dans l'ensemble des réels positifs. C'est « presque » une norme mais une propriété est manquante : la semi-norme d'un vecteur non nul peut être nulle. (fr)
rdfs:seeAlso	Normed space
name	Theorem (en)
dcterms:subject	Linear algebra Norms (mathematics)
Wikipage page ID	292067 (xsd:integer)
Wikipage revision ID	1122853396 (xsd:integer)
Link from a Wikipage to another Wikipage	Real-valued function Bounded set (topological vector space) Algebra over a field Homogeneous function Null vector Complex number Convex set Mathematics Norm (mathematics) Pullback Minkowski functional Convex function Convex hull Objective function Linear form Locally convex topological vector space Strong dual Sublinear function Composition algebra Functional analysis Balanced set Topological vector space Triangle inequality Weak-* topology Distributive lattice Hausdorff space Linear map Absolute value Absolutely convex set Linear algebra Nonnegative Normed space Isotropic quadratic form Kolmogorov's normability criterion Linear functional Quadratic form Pseudometric space Quotient space (linear algebra) Hahn–Banach theorem Involution (mathematics) Quasinorm Absorbing set Norms (mathematics) Positive definite Metrizable topological vector space Neighborhood basis Net (mathematics) Real number Locally convex Normed vector space T1 space Translation-invariant Vector subspace Indiscrete topology Reverse triangle inequality Positive homogeneous Balanced function F-seminorm Subadditive function
Link from a Wikipage to an external page	https://shodhganga.inflibnet.ac.in/bitstream/10603/13152/9/09_chapter%203.pdf https://arxiv.org/pdf/1611.02670.pdf
sameAs	Seminorm Seminorm Seminorm Seminorm Seminorm Seminorm Seminorm Seminorm Seminorm Seminorm
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