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Theorem stating that pointwise boundedness implies uniform boundedness
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with
Uniform_boundedness_principle
Property of functions
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is
Uniform_boundedness
Conjecture in number theory
algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the
Torsion_conjecture
Topics referred to by the same term
Uniform boundedness conjecture may refer to: Uniform boundedness conjecture for torsion points Uniform boundedness conjecture for rational points Uniform
Uniform boundedness conjecture
Uniform_boundedness_conjecture
Field of mathematics
in PN(K), and the general Uniform Boundedness Conjecture says that the number of preperiodic points in PN(K) may be bounded solely in terms of N, the
Arithmetic_dynamics
Area of mathematics
operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The
Functional_analysis
Mathematics conjecture about rational points on algebraic curves
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field K {\displaystyle K} and a positive
Uniform boundedness conjecture for rational points
Uniform_boundedness_conjecture_for_rational_points
Generalization of closed graph, open mapping, and uniform boundedness theorem
generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle. The following notation and notions are used, where R :
Ursescu_theorem
Two theorems about families of holomorphic functions
subset of the complex numbers is normal if and only if it is locally uniformly bounded. A family of holomorphic functions F {\displaystyle {\mathcal {F}}}
Montel's_theorem
Relation among continuous functions
holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between
Equicontinuity
Unsolved conjecture in geometry
Patricia Pacelli showed that the Bombieri–Lang conjecture implies a uniform boundedness conjecture for rational points: there is a constant B g , d {\displaystyle
Bombieri–Lang_conjecture
On topological spaces where the intersection of countably many dense open sets is dense
prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle. BCT1 also shows that every nonempty complete metric space
Baire_category_theorem
On convergent subsequences of functions that are locally of bounded total variation
a n } n = 1 ∞ {\displaystyle \{a_{n}\}_{n=1}^{\infty }} . By the uniform boundedness of { f n } n = 1 ∞ {\displaystyle \{f_{n}\}_{n=1}^{\infty }} and
Helly's_selection_theorem
implies χ ( R / P , R / Q ) > 0 {\displaystyle \chi (R/P,R/Q)>0} . Uniform boundedness conjecture for rational points: do algebraic curves of genus g ≥
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
On when a family of real, continuous functions has a uniformly convergent subsequence
satisfied by a uniformly bounded sequence {fn} of differentiable functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives
Arzelà–Ascoli_theorem
Topics referred to by the same term
in number theory Mazur's Conjecture B, a weaker variant of the uniform boundedness conjecture Mazur's lemma, a result in the theory of normed vector
Mazur's_theorem
American mathematics professor
of rational maps, and her recent proof (with DeMarco and Ye) of uniform boundedness results for numbers of torsion points on families of bielliptic genus
Holly_Krieger
Theorem on boundedness of symmetric operators
operators are closed. Alternatively, it can be argued using the uniform boundedness principle. One relies on the symmetric assumption, therefore the
Hellinger–Toeplitz_theorem
Theorem in measure theory
d\mu }=\int _{S}{f\,d\mu }.} Remark: The pointwise convergence and uniform boundedness of the sequence can be relaxed to hold only μ-almost everywhere,
Dominated_convergence_theorem
Decomposition of periodic functions
continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact. In
Fourier_series
Mathematical function whose set of values is bounded
a bounded set in Y {\displaystyle Y} .[citation needed] Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded
Bounded_function
Concept in mathematical analysis
lack of uniform integrability is behind many divergence phenomena for the Fourier series. For example, together with the uniform boundedness principle
Dirichlet_kernel
Mathematical concept
integrable functions and have uniformly bounded operator norms. Since the Riesz transforms are unitary on L2(C), the uniform boundedness of the truncated Riesz
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
Kind of linear transformation
implies that T is bounded, of course, but the converse need not be true. Another boundedness condition is that of polynomial boundedness: an operator T on
Bounded_operator
Generalization of boundedness
normability criterion – Characterization of normable spaces Local boundedness Totally bounded space – Generalization of compactness Narici & Beckenstein 2011
Bounded set (topological vector space)
Bounded_set_(topological_vector_space)
Mathematical problem in classical harmonic analysis
not uniformly. However, the Fourier series of a continuous function need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of
Convergence_of_Fourier_series
Type of topological vector space
Ultrabarrelled space Uniform boundedness principle#Generalisations – Theorem stating that pointwise boundedness implies uniform boundedness Ursescu theorem –
Barrelled_space
Continuous real function on a closed interval has a maximum and a minimum
[a,b].} The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function f {\displaystyle
Extreme_value_theorem
Austrian mathematician (1879–1934)
Hahn–Banach theorem and (independently of Banach and Steinhaus) the uniform boundedness principle. Other theorems include: the Hahn decomposition theorem;
Hans_Hahn_(mathematician)
under f {\displaystyle f} is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:
Local_boundedness
Mathematical concept
with L 1 {\displaystyle L^{1}} boundedness and tightness (conditions (1) and (3) in Theorem 3) is equivalent to uniform integrability. The following theorems
Uniform_integrability
Type of vector space in math
Every weakly convergent sequence {xn} is bounded, by the uniform boundedness principle. Conversely, every bounded sequence in a Hilbert space admits weakly
Hilbert_space
Concept in topology
is dense in X . {\displaystyle X.} A special case of this is the uniform boundedness principle. The empty space is a Baire space. It is the only space
Baire_space
Functional analysis concept
Zhu (2007, Theorem 1.14, p.11) and note in this reference that the uniform boundedness will apply in the situation where F ⊂ X {\displaystyle F\subset X}
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Condition for a linear operator to be open
Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Webbed space – Space where open mapping and closed graph
Open mapping theorem (functional analysis)
Open_mapping_theorem_(functional_analysis)
Generalization of topological interior
Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Aliprantis & Border 2006, pp. 199–200. John Cook (May 21
Algebraic_interior
Normed vector space that is complete
operators from X {\displaystyle X} to Y . {\displaystyle Y.} The uniform boundedness principle states that if for all x {\displaystyle x} in X {\displaystyle
Banach_space
Topics referred to by the same term
Publication Pueblo Memorial Airport, in Colorado, US The principle of uniform boundedness, in mathematics Pub rock (disambiguation) Public (disambiguation)
Pub_(disambiguation)
Type of convergence in Hilbert spaces
convex bounded closed set is weakly compact. As a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded. The
Weak convergence (Hilbert space)
Weak_convergence_(Hilbert_space)
Banach algebra, Functional analysis, Banach fixed-point theorem, uniform boundedness principle, Banach–Alaoglu theorem and Banach measure. Lwów School
Timeline of Polish science and technology
Timeline_of_Polish_science_and_technology
Generalization of compactness
mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered
Totally_bounded_space
Infrabarreled space Uniform boundedness principle#Generalisations – Theorem stating that pointwise boundedness implies uniform boundedness Khaleelulla 1982
Ultrabarrelled_space
Theorems connecting continuity to closure of graphs
Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Webbed space – Space where open mapping and closed graph
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Summability method used in harmonic analysis
equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular p ∈ ( 1 , ∞ ) {\displaystyle p\in (1
Bochner–Riesz_mean
Concept in applied mathematics
_{e_{3}}=\Gamma _{w_{4}}} Central differencing scheme satisfies first condition of boundedness. Since F e − F w = 0 {\displaystyle F_{e}-F_{w}=0} from continuity equation
Central_differencing_scheme
Theorem relating continuity to graphs
Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Webbed space – Space where open mapping and closed graph
Closed_graph_theorem
C\|\Delta _{1}u\|_{(k)}+C^{\prime }\|u\|_{(k+1)}.\end{aligned}}} The uniform boundedness of the difference quotients δhu implies that Yu lies in Hk+1(Ω) with
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Theorem in complex analysis
applying an appropriate version of the Fourier inversion theorem. The boundedness condition on φ ( s ) {\displaystyle \varphi (s)} can be strengthened
Mellin_inversion_theorem
Irish mathematician (1948–2006)
and elementary general functional analysis (Hahn–Banach theorem, uniform boundedness principle, Riesz-Kakutani theorem etc.). However, the theory of locally
Gerard_Murphy_(mathematician)
name convenient, which was borrowed from (Steenrod 1967). 7. Smooth uniform boundedness theorem ([KM], theorem 5.26). A linear mapping f : E → C ∞ ( V ,
Convenient_vector_space
Topics referred to by the same term
geometric criterion for local uniform convergence of univalent functions Borel–Carathéodory theorem, about the boundedness of a complex analytic function
Carathéodory's_theorem
Topological vector space whose topology can be defined by a metric
theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem In fact, this is true for topological group, for the proof doesn't
Metrizable topological vector space
Metrizable_topological_vector_space
Russian mathematician (1942–2022)
(2000). "Remarks about uniform boundedness of rational points over function fields". arXiv:math/0004078. Heier, Gordon (2003). "Uniformly effective Shafarevich
Aleksei_Parshin
Type of equation with integer coefficients
weaker form of a conjecture of Stewart, and is a special case of the uniform boundedness conjecture for rational points. This conjecture has been proven for
Thue_equation
Mathematical problem concerning limit cycles in dynamical systems
that persist under small perturbations. However, the question of uniform boundedness across parameter families remains meaningful and forms the basis
Hilbert–Arnold_problem
graph theorem Uniform boundedness principle Arzelà–Ascoli theorem Banach–Alaoglu theorem Measure of non-compactness Banach–Mazur theorem Bounded linear operator
List of functional analysis topics
List_of_functional_analysis_topics
operator, usually defined on a dense subspace. uniform boundedness principle The uniform boundedness principle states: given a set of operators between
Glossary of functional analysis
Glossary_of_functional_analysis
Representation theory of the symplectic group
z ) = W ( z ) u {\displaystyle \Phi (z)=W(z)u} is smooth. By the uniform boundedness theorem, this is equivalent to the requirement that each matrix coefficient
Oscillator_representation
models includes the SETAR model as a limiting case by showing the uniform boundedness and equicontinuity with respect to the switching parameter. Without
STAR_model
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
^{2}+t^{2}}\leq {2|\lambda | \over \lambda ^{2}+t^{2}}+C_{1}.}} Uniform boundedness follows because the first term has a finite integral independent
Neumann–Poincaré_operator
Number of independent rational basis points with infinite order
rank on average as well. In Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Bhargava and Shankar
Rank_of_an_elliptic_curve
such that f(gh) − f(g) − f(h) is uniformly bounded and f(gn) = n·f(g). The vector space of pseudocharacters of a boundedly generated group G is finite-dimensional
Boundedly_generated_group
Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Zălinescu 2002, pp. 1–23. Zălinescu, Constantin (30 July
Convex_series
Space where open mapping and closed graph theorems hold
Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Narici & Beckenstein 2011, p. 470−471. Narici & Beckenstein
Webbed_space
Countably quasi-barrelled space Infrabarreled space Ultrabarrelled space Uniform boundedness principle#Generalisations Khaleelulla 1982, pp. 65–76. Bourbaki,
Quasi-ultrabarrelled_space
Complexity class used in circuit complexity
bounded by functions of FP and ( y -th bit of f ( x 1 , … , x n ) ) {\displaystyle (y{\text{-th bit of }}f(x_{1},\ldots ,x_{n}))} is in the uniform TC0
TC0
Continuous maps on a closed subset of a normal space can be extended
normal topological space can be extended to the entire space, preserving boundedness if necessary. If X {\displaystyle X} is a normal space and f : A → R
Tietze_extension_theorem
State of being limited or ended
Middle Ages through Old French and Middle English, initially referring to boundedness in space or quantity, and later acquiring broader abstract uses. In mathematics
Finiteness
Concept in geometry and topology
neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features
Coarse_structure
Theorems concerning stochastic processes
typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random
Doob's martingale convergence theorems
Doob's_martingale_convergence_theorems
on F {\displaystyle F} is Hausdorff. Boundedness A subset H {\displaystyle H} of F {\displaystyle F} is bounded in the G {\displaystyle {\mathcal {G}}}
Topologies on spaces of linear maps
Topologies_on_spaces_of_linear_maps
French mathematician
4171/RMI/17 with Jean-Lin Journé: David, Guy; Journé, Jean-Lin (1984), "A boundedness criterion for generalized Calderón-Zygmund operators", Annals of Mathematics
Guy_David_(mathematician)
Mathematical theorem
be a totally bounded sequence and chose a countable dense subset w m {\displaystyle w_{m}} of G {\displaystyle G} . By locally boundedness and a "diagonal
Riemann_mapping_theorem
In control theory, visible state of a system
S2CID 51615852. Li, W.; Wang, Z.; Ho, D. W. C.; Wei, G. (2019). "On Boundedness of Error Covariances for Kalman Consensus Filtering Problems". IEEE Transactions
Observability
Type of continuous linear operator
functions satisfying uniform L p {\displaystyle L^{p}} -type bounds on translated real slices. Interior estimates again imply local boundedness on compact subsets
Compact_operator
Mathematical technique in complex analysis
technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function f {\displaystyle f} (i.e., | f ( z ) | < M
Phragmén–Lindelöf_principle
Mathematical function often applied to matrices
M[0]=0,\quad M[I]=1} M [ − A ] = − m [ A ] {\displaystyle M[-A]=-m[A]} Boundedness − ‖ A ‖ ≤ m [ A ] ≤ M [ A ] ≤ ‖ A ‖ {\displaystyle \,-\|A\|\,\leq \,m[A]\
Logarithmic_norm
Function between topological vector spaces
is bounded. Function bounded on a neighborhood and local boundedness In contrast, a map F : X → Y {\displaystyle F:X\to Y} is said to be bounded on a
Continuous_linear_operator
Vector space with a notion of nearness
definition of boundedness can be weakened a bit; E {\displaystyle E} is bounded if and only if every countable subset of it is bounded. A set is bounded if and
Topological_vector_space
Theorem
conditions. As in the pointwise case of the Jordan test, the condition of boundedness can be relaxed if the function is assumed to be absolutely integrable
Dirichlet–Jordan_test
plane C. Melnikov and Verdera (1995) showed the precise relation of the boundedness of the Cauchy kernel to the curvature of measures. They proved that if
Curvature_of_a_measure
Field in logic and theoretical computer science
fragments of Peano arithmetic, which come under the name of bounded arithmetic, serve as uniform versions of propositional proof systems and provide further
Proof_complexity
Computational tool
consequence of the boundedness on the space Lp([0, 2π]) of the Hilbert transform on the circle. It follows from this boundedness that the projections
Schauder_basis
Grammatical category for verbs
there are two theories about passive voice in Japanese called the uniform and non-uniform theory.[citation needed] These two theories debate whether direct
Voice_(grammar)
Theorem in statistics
In statistics, Basu's theorem states that any boundedly complete and sufficient statistic is independent of any ancillary statistic. This is a 1955 result
Basu's_theorem
Topological space with a bounded image under any continuous function to R
Topological Groups: Between Compactness and ℵ 0 {\displaystyle \aleph _{0}} -boundedness, in Mirek Husek and Jan van Mill (eds.), Recent Progress in General Topology
Pseudocompact_space
Problem in mathematical sociology
amount, ∀ i : x i ≥ 0 {\displaystyle \forall i:x_{i}\geq 0} . Claims-boundedness: each claimant should get at most his claim, ∀ i : x i ≤ c i {\displaystyle
Bankruptcy_problem
Dual space topology of uniform convergence on some sub-collection of bounded subsets
topology of G {\displaystyle {\mathcal {G}}} -convergence or topology of uniform convergence on the sets of G {\displaystyle {\mathcal {G}}} is a method
Polar_topology
Expression of time reference in grammar
Lexical aspect (Aktionsart) Mood Tense Voice General features Affect Boundedness Comparison (degree) Egophoricity Pluractionality (verbal number) Honorifics
Grammatical_tense
1966 result in mathematical analysis
theorem follows from the boundedness of the Carleson operator from Lp(R) to itself for 1 < p < ∞. However, proving that it is bounded is difficult, and this
Carleson's_theorem
and is bounded by 1. Cotlar 1955 Stein 1993 Hörmander 1994 Knapp & Stein 1971 Calderon, Alberto; Vaillancourt, Remi (1971). "On the boundedness of pseudo-differential
Cotlar–Stein_lemma
Mathematical method in extremal graph theory
given k {\displaystyle k} -uniform hypergraph into a random-like object with bounded parts (with an appropriate boundedness and randomness notions) that
Hypergraph_regularity_method
Theorem in topology
three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge." He also noted that the search
Brouwer_fixed-point_theorem
Mathematics term
positive definite functions on G converging to 1 uniformly on compact subsets, converges to 1 uniformly on G. (3) Every unitary representation of G that
Kazhdan's_property_(T)
this special property is called boundedness. A fundamental property of Fano varieties is that they fail to be bounded, and thus their stability cannot
K-stability_of_Fano_varieties
Statistics term
is said to be boundedly complete for the distribution of X if this implication holds for every measurable function g that is also bounded. The Bernoulli
Completeness_(statistics)
Type of random mathematical object
location will be a uniform random variable defined on that interval. Furthermore, the homogeneous point process is sometimes called the uniform Poisson point
Poisson_point_process
Mathematical functions which are smooth but not analytic
}}\end{cases}}\quad k,n\in \mathbb {N} _{0},} and the boundedness theorem implies that ψn and every derivative of ψn is bounded. Therefore, the constants λ n = max {
Non-analytic_smooth_function
Space where bounded operators are continuous
possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses
Bornological_space
Linear operator equal to its own adjoint
\mathbb {C} \setminus [m,M].} The goal is to prove the existence and boundedness of R λ − 1 , {\displaystyle R_{\lambda }^{-1},} and show that Dom R
Self-adjoint_operator
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