functional analysis, a seminorm is a norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski...

quasi-seminorm or a Banach functional, on a vector space X {\displaystyle X} is a real-valued function with only some of the properties of a seminorm. Unlike...

with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A...

multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm). The sum of finitely many F-seminorms (resp. F-norms)...

Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general...

Grothendieck. The topology on nuclear spaces can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements...

|F|\leq p.} Every norm is a seminorm and both are symmetric balanced sublinear functions. A sublinear function is a seminorm if and only if it is a balanced...

of that integral. The problem is that this is not really a norm, only a seminorm, because there are functions other than the zero function whose (semi)norms...

with a norm. A seminormed vector space is a vector space equipped with a seminorm. A useful variation of the triangle inequality is ‖ x − y ‖ ≥ | ‖ x ‖ −...

that it gives rise to a seminorm rather than a vector space norm. The quotient of this space by the kernel of this seminorm is also required to be a...

quasi-seminorm) whose multiplier is 1. {\displaystyle 1.} Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm). If...

K {\displaystyle p_{K}} will be a real-valued seminorm on X . {\displaystyle X.} In fact, every seminorm p {\displaystyle p} on X {\displaystyle X} is...

is complete with respect to the family of seminorms. A family P {\displaystyle {\mathcal {P}}} of seminorms on X {\displaystyle X} yields a Hausdorff...

generalizes the definition of seminorms since a map is a seminorm if and only if it is a 1 {\displaystyle 1} -seminorm (using p := 1 {\displaystyle p:=1}...

positive definiteness is omitted, then p {\displaystyle p} is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that...

non-negativity are the defining properties of a seminorm. Thus ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} is a seminorm and the set L p ( S , μ ) {\displaystyle...

with exponent α in Ω. In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of Ω, then...

null vector, a linear mapping given as matrix product or dot product, a seminorm in a Minkowski space, etc.). In set theory, the empty set, that is, the...

include: for every continuous seminorm q {\displaystyle q} on Y , {\displaystyle Y,} there exists a continuous seminorm p {\displaystyle p} on X {\displaystyle...

this linear functional f {\displaystyle f} is dominated by some given seminorm p : X → R {\displaystyle p:X\to \mathbb {R} } (meaning that | f ( m ) |...

then the weak topology 𝜎(X, Y, b) on X is induced by the family of seminorms, py : X → R {\displaystyle \mathbb {R} } , defined by py(x) := |b(x, y)|...

"size" of linear operators Operator theory – Mathematical field of study Seminorm – Mathematical function Unbounded operator – Linear operator defined on...

{\displaystyle u\in L^{p}(\Omega )} and the seminorm [ u ] s , p {\displaystyle [u]_{s,p}} is finite. The seminorm [ u ] s , p {\displaystyle [u]_{s,p}} is...

Hölder continuous functions over the domain Ω {\displaystyle \Omega } . The seminorm of the Morrey spaces is given by ( [ u ] λ , p ) p = sup 0 < r < diam ⁡...

functions. It is the closure of the trigonometric polynomials under the seminorm ‖ f ‖ W , p = lim r → ∞ ‖ f ‖ S , r , p {\displaystyle \|f\|_{W,p}=\lim...

space, hence a topological group). But it will not be Tychonoff if the seminorm is not a norm. Generalizing both the metric spaces and the topological...

numbers as well as over the complex numbers. More generally, every norm and seminorm is a positively homogeneous function of degree 1 which is not a homogeneous...

identify functions that agree almost everywhere to get a norm, and not only a seminorm. "Many functions in L 2 {\displaystyle L^{2}} of Lebesgue measure, being...

enormous, nonnormal, nonnormative, norm, normal, normality, normative, seminorm, seminormal, subnormal noster nostr- our nostrum novem novem- nine November...

convex space if and only if the topology of the TVS is induced by some seminorm. In particular, every locally bounded TVS is pseudometrizable. Let f :...