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...

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...

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...

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...

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...

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 ) |...

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...

"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 ⁡...

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)|...

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...

for f , g ∈ F ( X ) {\displaystyle f,g\in {\mathcal {F}}(X)} A seminorm p {\displaystyle p} induces the pseudometric d ( x , y ) = p ( x − y )...

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