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Mathematical function
analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski
Seminorm
Length in a vector space
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". It
Norm_(mathematics)
Theorem on extension of bounded linear functionals
|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
Hahn–Banach_theorem
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
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
Nuclear_space
Space with topology generated by convex sets
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
Locally convex topological vector space
Locally_convex_topological_vector_space
Type of function in linear algebra
also called a quasi-seminorm, on a vector space is a real-valued function with some of the properties of a seminorm. Unlike seminorms, a sublinear function
Sublinear_function
Type of function in linear algebra
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
Quasinorm
Vector space on which a distance is defined
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 ‖ −
Normed_vector_space
Topological vector space whose topology can be defined by a metric
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)
Metrizable topological vector space
Metrizable_topological_vector_space
Inequality between integrals in Lp spaces
the space of all complex-valued functions on S. Let N be an increasing seminorm on F ( S , C ) , {\displaystyle F(S,\mathbb {C} ),} meaning that, for all
Hölder's_inequality
Locally convex topological vector space that is also a complete metric space
is complete with respect to the family of seminorms. A family P {\displaystyle {\mathcal {P}}} of seminorms on X {\displaystyle X} yields a Hausdorff
Fréchet_space
Generalization of the concept of a norm
positive definiteness is omitted, then p {\displaystyle p} is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that
Asymmetric_norm
Normed vector space that is complete
if and only if the seminorm | f | {\displaystyle |f|} is continuous, which happens if and only if there exists a continuous seminorm p : X → R {\displaystyle
Banach_space
Function made from a set
guarantee that p K {\textstyle p_{K}} will be a seminorm on X . {\textstyle X.} In fact, every seminorm p {\textstyle p} on X {\textstyle X} is equal to
Minkowski_functional
Type of continuity of a complex-valued function
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
Hölder_condition
Function spaces generalizing finite-dimensional p norm spaces
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
Lp_space
Mathematical representation of absence of a value
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
Null_(mathematics)
Concept in topology
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
Kolmogorov_space
Convex and balanced set
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}
Absolutely_convex_set
Mathematical concept
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
Semi-Hilbert_space
Analytic space in mathematics
value is the corresponding seminorm in the Berkovich spectrum. Ostrowski's theorem shows that any multiplicative seminorm on the integers (with the usual
Berkovich_space
{\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }} is an increasing family of seminorms for the topology of A {\displaystyle A} , the joint continuity of multiplication
Fréchet_algebra
X ′ {\displaystyle X'} . A seminorm p {\displaystyle p} on X {\displaystyle X} is called Hilbertian or a Hilbert seminorm if there exists a positive definite
Minlos–Sazonov_theorem
Concept in fluid dynamics
j=1}^{n}\left|\partial _{i}u^{j}\right|^{2}} . This quantity is the squared seminorm | u | H 1 ( Ω ) n 2 {\displaystyle |\mathbf {u} |_{H^{1}(\Omega )^{n}}^{2}}
Enstrophy
Mathematical function, in linear algebra
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 ) |
Linear_map
nonnormative, norm, normable, normal, normality, normative, quasinorm, seminorm, seminormable, seminormal, subnormal not- south Greek νότος (nótos) Notogaea
List of Greek and Latin roots in English/H–O
List_of_Greek_and_Latin_roots_in_English/H–O
Algebraic structure in linear algebra
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
Vector_space
Objects that generalize functions
)<\infty .} The family of seminorms pα, β defines a locally convex topology on the Schwartz space. When n is equal to 1, the seminorms are, in fact, norms on
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
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
Morrey–Campanato_space
Type of metric space
all complex sequences for which it is finite. (Note that this is not a seminorm since it lacks homogeneity — If the r n {\displaystyle r_{n}} are allowed
Ultrametric_space
Mathematical concept
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)|
Weak_topology
Vector space with generalized dot product
V.} Ptolemy's inequality is a necessary and sufficient condition for a seminorm to be the norm defined by an inner product. Orthogonality Two vectors x
Inner_product_space
Function with a multiplicative scaling behaviour
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
Homogeneous_function
Infimum and supremum almost everywhere
functions that are bounded almost everywhere is a seminormed space whose seminorm ‖ f ‖ ∞ = inf { C ∈ R ≥ 0 : | f ( x ) | ≤ C for almost every x } = {
Essential infimum and essential supremum
Essential_infimum_and_essential_supremum
Type of regular Hausdorff space
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
Tychonoff_space
Alternative representations of the RGB color model
written as max(R, G, B) − min(R, G, B), and showing that this value is a seminorm. They reserve the name chroma for the Euclidean norm in the chromaticity
HSL_and_HSV
Generalization of boundedness
is defined by a family P {\displaystyle {\mathcal {P}}} of continuous seminorms, then this list may be extended to include: p ( B ) {\displaystyle p(B)}
Bounded set (topological vector space)
Bounded_set_(topological_vector_space)
Mathematical inequality in Sobolev space theory
u} such that u ∈ L p ( Ω ) {\displaystyle u\in L^{p}(\Omega )} and the seminorm [ u ] s , p {\displaystyle [u]_{s,p}} is finite, where [ u ] s , p {\displaystyle
Poincaré_inequality
Distance function defined between probability distributions
may define for f : M → R {\displaystyle f\colon M\to \mathbb {R} } the seminorm ‖ f ‖ H ˙ 1 ( π ) 2 = ∫ M ‖ ∇ f ( x ) ‖ 2 π ( d x ) {\displaystyle \|f\|_{{\dot
Wasserstein_metric
In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1
P-variation
Topologies on operators on a Hilbert space
all locally convex, which implies that they are defined by a family of seminorms. In analysis, a topology is called strong if it has many open sets and
Operator_topologies
operators on a Hilbert space is the topology defined by the family of seminorms p ω ( x ) = ω ( x ∗ x ) 1 / 2 {\displaystyle p_{\omega }(x)=\omega (x^{*}x)^{1/2}}
Ultrastrong_topology
Concept in mathematics
seminormed space, that is a vector space with the topology induced by a seminorm, all neighbourhood systems can be constructed by translation of the neighbourhood
Neighbourhood_system
enormous, nonnormal, nonnormative, norm, normal, normality, normative, seminorm, seminormal, subnormal noster nostr- our nostrum novem novem- nine November
List of Latin words with English derivatives
List_of_Latin_words_with_English_derivatives
Euclidean Wightman distributions
constant, | f | C ⋅ n {\displaystyle |f|_{C\cdot n}} is the Schwartz-space seminorm of order N = C ⋅ n {\displaystyle N=C\cdot n} , i.e. | f | N = sup | α
Schwinger_function
{\displaystyle Y} are induced by seminorms, the topology of X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is induced by seminorms constructed from those on X {\displaystyle
Projective_tensor_product
Characterization of surjectivity
X^{\prime }.} For every continuous seminorm p {\displaystyle p} on X {\displaystyle X} there exists a continuous seminorm q {\displaystyle q} on Y {\displaystyle
Surjection_of_Fréchet_spaces
Function which is integrable on its domain
^{+}} , k ∈ N {\displaystyle k\in \mathbb {N} } is an indexed family of seminorms, defined as ‖ u ‖ p , ω k = ( ∫ ω k | u ( x ) | p d x ) 1 / p ∀ u ∈ L
Locally_integrable_function
Vector space with a notion of nearness
boundedness can be characterized by seminorms: the subset E {\displaystyle E} is bounded if and only if every continuous seminorm p {\displaystyle p} is bounded
Topological_vector_space
Topological space with a notion of uniform properties
particularly useful in functional analysis (with pseudometrics provided by seminorms). More precisely, let f : X × X → R {\displaystyle f:X\times X\to \mathbb
Uniform_space
Function that "converges" to periodicity
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
Almost_periodic_function
Topics referred to by the same term
measured as an angle p-variation in mathematical analysis, a family of seminorms of functions Coefficient of variation in probability theory and statistics
Variation
Vector space of infinite sequences
{\displaystyle X} admits no continuous norm (that is, any continuous seminorm on X {\displaystyle X} has a nontrivial null space). X {\displaystyle
Sequence_space
Function between topological vector spaces
include: for every continuous seminorm q {\displaystyle q} on Y , {\displaystyle Y,} there exists a continuous seminorm p {\displaystyle p} on X {\displaystyle
Continuous_linear_operator
Generalization of metric spaces in mathematics
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 )
Pseudometric_space
Theorem about inclusions between Sobolev spaces
(It should as well be stressed that on the left hand side is the BMO seminorm.) The prior result may be extended to arbitrary degree k {\displaystyle
Sobolev_inequality
Degree of differentiability of a function or map
Fréchet space. One way to describe this topology is by the family of seminorms p K , α ( f ) = sup x ∈ K | D α f ( x ) | , {\displaystyle p_{K,\alpha
Smoothness
Function reducing distance between all points
point. In a locally convex space (E, P) with topology given by a set P of seminorms, one can define for any p ∈ P a p-contraction as a map f such that there
Contraction_mapping
Mode of convergence of a function sequence
compact, σ-compact Hausdorff space, this topology is generated by the seminorms p K ( f ) = sup x ∈ K | f ( x ) | , {\displaystyle p_{K}(f)=\sup _{x\in
Uniform_convergence
Infinite sum
It is called absolutely summable if in addition, for every continuous seminorm p {\displaystyle p} on X , {\displaystyle X,} the family ( p ( x i ) )
Series_(mathematics)
nonnormative, norm, normable, normal, normality, normative, quasinorm, seminorm, seminormable, seminormal, subnormal not- south Greek νότος (nótos) Notogaea
List of Greek and Latin roots in English/N
List_of_Greek_and_Latin_roots_in_English/N
Complex-differentiable (mathematical) function
connected. In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets. From a geometric perspective, a
Holomorphic_function
Auxiliary functions used to probe equations, distributions, and weak formulations
)<\infty .} The family of seminorms pα, β defines a locally convex topology on the Schwartz space. When n is equal to 1, the seminorms are, in fact, norms on
Test_function
Relation among continuous functions
include: for every continuous seminorm q {\displaystyle q} on Y , {\displaystyle Y,} there exists a continuous seminorm p {\displaystyle p} on X {\displaystyle
Equicontinuity
Vector space of functions in mathematics
{\displaystyle f\in L^{p}(\Omega ),} the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by [ f ] θ , p , Ω := ( ∫ Ω ∫ Ω | f (
Sobolev_space
Mathematical space with a notion of distance
is a norm induced by the metric. A similar relationship holds between seminorms and pseudometrics. Among examples of metrics induced by a norm are the
Metric_space
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 :
Local_boundedness
Kind of linear transformation
linear operators Operator theory – Mathematical study of linear operators Seminorm – Mathematical function Unbounded operator – Linear operator defined on
Bounded_operator
of metric differentials holds: for almost every z in Rn, MD(f, z) is a seminorm and d X ( f ( x ) , f ( y ) ) − M D ( f , z ) ( x − y ) = o ( | x − z |
Metric_differential
List of definitions of terms and concepts commonly used in aerospace engineering
_{i}u^{j}\right|^{2}} . This is quantity is the same as the squared seminorm | u | H 1 ( Ω ) n 2 {\displaystyle |\mathbf {u} |_{H^{1}(\Omega )^{n}}^{2}}
Glossary of aerospace engineering
Glossary_of_aerospace_engineering
Linear map from a vector space to its field of scalars
vector space X is continuous if and only if there exists a continuous seminorm p on X such that | f | ≤ p . {\displaystyle |f|\leq p.} Continuous linear
Linear_form
Weak topology on function spaces
{\displaystyle x\in X} and y ∗ ∈ Y ∗ {\displaystyle y^{*}\in Y^{*}} defines a seminorm ‖ ⋅ ‖ x , y ∗ {\displaystyle \|\cdot \|_{x,y^{*}}} on B ( X , Y ) {\displaystyle
Weak_operator_topology
Collection of results for partial differential equations
α ( Ω ) {\displaystyle f\in C^{0,\alpha }(\Omega )} , the usual Hölder seminorm is given by [ f ] 0 , α ; Ω = sup x , y ∈ Ω | f ( x ) − f ( y ) | | x −
Schauder_estimates
Concept in mathematics
{\displaystyle (s_{j})_{j\in J}} of simple functions such that for every continuous seminorm p {\displaystyle p} on E {\displaystyle E} ∫ X p ( f − s j ) d μ → 0 {\displaystyle
Bochner_integral
Set that can be "inflated" to reach any point
{\displaystyle p_{D}:X\to \mathbb {R} } of D {\displaystyle D} will be a seminorm on X , {\displaystyle X,} thereby making ( X , p D ) {\displaystyle \left(X
Absorbing_set
Method for assigning values to integrals
{C_{c}^{\infty }}(\mathbb {R} )\to \mathbb {C} } is bounded by the usual seminorms for Schwartz functions u {\displaystyle u} . Therefore, this map defines
Cauchy_principal_value
Locally convex topology on function spaces
topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form T ↦ ‖ T x ‖ {\displaystyle T\mapsto \|Tx\|} , as x varies
Strong_operator_topology
In mathematics, vector space of linear forms
{\mathcal {A}},} or what is the same thing, the topology generated by seminorms of the form ‖ φ ‖ A = sup x ∈ A | φ ( x ) | , {\displaystyle \|\varphi
Dual_space
Class of norms in additive combinatorics
Terence; Ziegler, Tamar (2010). "An inverse theorem for the uniformity seminorms associated with the action of F p ∞ {\displaystyle \mathbb {F} _{p}^{\infty
Gowers_norm
\rho {\text{ is a representation of }}(G,R)\}} is finite and defines a seminorm satisfying the C*-norm condition on the free algebra on X. The completion
Universal_C*-algebra
Subset of all points that is bounded by some given point of a dual (in a dual pairing)
\left|x^{\prime }(A)\right|~:=~\sup _{a\in A}\left|x^{\prime }(a)\right|} is a seminorm on Y . {\displaystyle Y.} If A = ∅ {\displaystyle A=\varnothing } then
Polar_set
values in a Fréchet space X {\displaystyle {\mathcal {X}}} (equipped with seminorms || ⋅ ||α). This includes most common settings of vector-valued random
Multidimensional Chebyshev's inequality
Multidimensional_Chebyshev's_inequality
{\displaystyle H_{2}(M,\mathbb {Q} )} which can then be extended by continuity to a seminorm ‖ ⋅ ‖ T {\displaystyle \|\cdot \|_{T}} on H 2 ( M , R ) {\displaystyle
Thurston_norm
Theorem in functional analysis
{\scriptscriptstyle {\text{def}}}{=}}~\left(m_{x}\right)_{x\in X}:X\to [0,\infty )} is a seminorm and it is unchanged if U {\displaystyle U} is replaced by the convex balanced
Banach–Alaoglu_theorem
on spectral triples (and more generally, on seminorms which play a role of analogue for Lipschitz seminorms) for Connes' distance to indeed induce the
Spectral_triple
well-defined and forms a seminorm on span D . {\displaystyle \operatorname {span} D.} The locally convex topology induced by this seminorm is the topology τ
Auxiliary_normed_space
Construct in functional analysis
in a normed vector space are balanced sets. If p {\displaystyle p} is a seminorm (or norm) on a vector space X {\displaystyle X} then for any constant c
Balanced_set
Barrelled space where closed and bounded subsets are compact
is a Montel space equipped with the topology induced by the family of seminorms ‖ f ‖ K , n = sup | α | ≤ n sup x ∈ K | ∂ α f ( x ) | {\displaystyle \|f\|_{K
Montel_space
Common point(s) shared by two lines in Euclidean geometry
except for a zero eigenvalue in the direction along the line providing a seminorm on the distance between pi and another point giving the distance to the
Line–line_intersection
French mathematician
as listed in the 2025 preprint "On the relation between distances and seminorms on Fréchet spaces, with application to isometries", with Lucas Oger and
Isabelle_Chalendar
Mathematical function often applied to matrices
operator norms. The least upper bound Lipschitz constant is an operator seminorm, which is all that is required. A corresponding lower Lipschitz constant
Logarithmic_norm
Vector space consisting of affine subsets
is generated by a family of seminorms {pα | α ∈ A} where A is an index set. Let M be a closed subspace, and define seminorms qα on X/M by q α ( [ x ] )
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Type of topology
to Y. The compact-open topology is the initial topology induced by the seminorms p K ( f ) = sup { ‖ D j f ( x ) ‖ : x ∈ K , 0 ≤ j ≤ m } {\displaystyle
Compact-open_topology
Mathematical objects that generalise the notion of Hilbert spaces
Hausdorff completion of E ⊙ F {\displaystyle E\odot F} in the resulting seminorm is denoted E ⊗ B F {\displaystyle E\otimes _{B}F} . The left- and right-actions
Hilbert_C*-module
respect to x 2 {\displaystyle \textstyle x_{2}} , and so on. The Sobolev seminorm on W p m ( Ω ) {\displaystyle \textstyle W_{p}^{m}(\Omega )} consists of
Bramble–Hilbert_lemma
Product of any collection of compact topological spaces is compact
f} on a subspace V 0 ⊂ V {\displaystyle V_{0}\subset V} dominated by a seminorm p {\displaystyle p} , consider X {\displaystyle X} = the set of all finite-dimensional
Tychonoff's_theorem
Generalization of the inverse function theorem
following data: a vector space F {\displaystyle F} a countable collection of seminorms ‖ ⋅ ‖ n : F → R {\displaystyle \|\,\cdot \,\|_{n}:F\to \mathbb {R} } such
Nash–Moser_theorem
On surjectivity of linear map to anti-dual
space. If B {\displaystyle B} is non-negative then it induces a canonical seminorm on H {\displaystyle H} , denoted by ‖ ⋅ ‖ {\displaystyle \|\cdot \|} ,
Fundamental theorem of Hilbert spaces
Fundamental_theorem_of_Hilbert_spaces
Mode of convergence of an infinite series
i {\textstyle x_{H}:=\sum _{i\in H}x_{i}} ), and for every continuous seminorm p {\displaystyle p} on X , {\displaystyle X,} the family ( p ( x α ) )
Absolute_convergence
\|^{\infty }} is a seminorm on B ( X , Ω ) , {\displaystyle {\mathcal {B}}(X,\Omega ),} but not necessarily a norm. The kernel of this seminorm, N ∞ := { f ∈
Spectral theory of normal C*-algebras
Spectral_theory_of_normal_C*-algebras
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