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Mathematical theorem
the Heine–Cantor theorem states that a continuous function between two metric spaces is uniformly continuous if its domain is compact. The theorem is named
Heine–Cantor_theorem
Subset of Euclidean space is compact if and only if it is closed and bounded
In real analysis in mathematics, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S {\displaystyle S} of Euclidean
Heine–Borel_theorem
Mathematical theorem relating to limits
Heine's theorem, named after the German mathematician Eduard Heine, establishes a link in mathematical analysis between limits of functions and limits
Heine_theorem
German mathematician (1821–1881)
Andréief–Heine identity Heine theorem Heine–Borel theorem Heine–Cantor theorem Heine definition of continuity Heine's Reciprocal Square Root Identity Heine–Stieltjes
Eduard_Heine
Andréief–Heine identity Heine–Borel theorem Heine–Cantor theorem Heine–Stieltjes polynomials Heine definition of continuity Heine functions Heine's identity
List of things named after Eduard Heine
List_of_things_named_after_Eduard_Heine
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
closed and bounded subsets. This form of the theorem makes especially clear the analogy to the Heine–Borel theorem, which asserts that a subset of R n {\displaystyle
Bolzano–Weierstrass_theorem
Type of mathematical space
space, compactness is equivalent to being closed and bounded, by the Heine–Borel theorem. The property of compactness often allows local information to be
Compact_space
Probability theorem
theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine's
Continuous_mapping_theorem
Relates three different kinds of weak compactness in a Banach space
point in A. Compactness (or Heine-Borel compactness): Every open cover of A admits a finite subcover. The Eberlein–Šmulian theorem states that the three are
Eberlein–Šmulian_theorem
Property of a partially ordered set
as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme value theorem, and the Heine–Borel theorem. It is usually taken as an
Least-upper-bound_property
review. Heine–Borel theorem. This theorem was proved in 1872 by Émile Borel, not by Eduard Heine. Borel used techniques similar to those that Heine used
List_of_misnamed_theorems
rank Cantor–Bendixson theorem Cantor–Bernstein theorem Cantor–Dedekind axiom Heine–Cantor theorem Cantor–Schröder–Bernstein theorem Cantor–Schröder–Bernstein
List of things named after Georg Cantor
List_of_things_named_after_Georg_Cantor
Continuous real function on a closed interval has a maximum and a minimum
"every open cover of K {\displaystyle K} has a finite subcover". The Heine–Borel theorem asserts that a subset of the real line is compact if and only if
Extreme_value_theorem
Topics referred to by the same term
non-empty intersection Heine–Cantor theorem: a continuous function on a compact space is uniformly continuous Cantor–Bendixson theorem: a closed set of a
Cantor's theorem (disambiguation)
Cantor's_theorem_(disambiguation)
Modern application of infinitesimals
on a compact interval I is necessarily uniformly continuous (the Heine–Cantor theorem) admits a succinct hyperreal proof. Let x, y be hyperreals in the
Nonstandard_calculus
On when a family of real, continuous functions has a uniformly convergent subsequence
forms an open cover of I. Since I is closed and bounded, by the Heine–Borel theorem I is compact, implying that this covering admits a finite subcover
Arzelà–Ascoli_theorem
On decreasing nested sequences of non-empty compact sets
version follows from the general topological statement in light of the Heine–Borel theorem, which states that sets of real numbers are compact if and only if
Cantor's_intersection_theorem
Hahn–Mazurkiewicz theorem (continuum theory) Heine–Borel theorem (real analysis) Heine–Cantor theorem (metric geometry) Jordan curve theorem (topology) Kuratowski's
List_of_theorems
Q-analog of hypergeometric series
_{1}(q^{\alpha },q^{\beta };q^{\gamma };q,x)} was first considered by Eduard Heine (1846). It becomes the hypergeometric series F ( α , β ; γ ; x ) {\displaystyle
Basic_hypergeometric_series
In mathematics, F. Riesz's theorem (named after Frigyes Riesz) is a theorem in functional analysis that states that a Hausdorff topological vector space
F._Riesz's_theorem
German mathematician (1805–1859)
mathematician. In number theory, he proved special cases of Fermat's Last Theorem and created analytic number theory. In analysis, he advanced the theory
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
theorem Goodstein's theorem Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's
List_of_mathematical_proofs
Branch of mathematical logic
equivalent to weak Kőnig's lemma and thus to WKL0 over RCA0: The Heine–Borel theorem for the closed unit real interval, in the following sense: every
Reverse_mathematics
Mathematical theorem named after Pierre Cousin
student of Henri Poincaré, in 1895, and it extends the original Heine–Borel theorem on compactness for arbitrary covers of compact subsets of R n {\displaystyle
Cousin's_theorem
First article on transfinite set theory
Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary
Cantor's first set theory article
Cantor's_first_set_theory_article
Mathematician (1845–1918)
more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal
Georg_Cantor
interval is also an interval Heine–Borel theorem – sometimes used as the defining property of compactness Bolzano–Weierstrass theorem – states that each bounded
List_of_real_analysis_topics
Mathematics of real numbers and real functions
Heine–Borel theorems, L'Hopital's rule, the mean value theorem, Taylor's theorem, the fundamental theorem of calculus, and the extreme value theorem.
Real_analysis
Theorem in functional analysis
and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of
Banach–Alaoglu_theorem
Book by John Stillwell
extreme value theorem, the Heine–Cantor theorem on uniform continuity, the Hahn–Banach theorem, and the Riemann mapping theorem. These theorems are analyzed
Reverse Mathematics: Proofs from the Inside Out
Reverse_Mathematics:_Proofs_from_the_Inside_Out
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
the completion of the space is compact). This is analogous to the Heine-Borel theorem. In contrast, no infinite-dimensional normed space has this property
Nuclear_space
German mathematician
Fyodorov–Schoenflies–Bieberbach theorem Jordan–Schoenflies theorem Schoenflies notation Schoenflies displacement Heine–Borel theorem Geometrical crystallography
Arthur_Moritz_Schoenflies
Metric geometry
is complete and totally bounded. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace S {\displaystyle
Complete_metric_space
category theorem Nowhere dense Baire space Banach–Mazur game Meagre set Comeagre set Compact space Relatively compact subspace Heine–Borel theorem Tychonoff's
List of general topology topics
List_of_general_topology_topics
View that mathematics does not necessarily represent reality, but is more akin to a game
German mathematicians Eduard Heine and Carl Johannes Thomae are considered early advocates of mathematical formalism. Heine and Thomae's formalism can be
Formalism (philosophy of mathematics)
Formalism_(philosophy_of_mathematics)
Term meaning 'not', 'without', or 'lack'
responsible for this peculiarity. A similar critique has been given by Steven Heine: The common approach espoused [...] emphasizes a particular understanding
Mu_(negative)
Differentiation under the integral sign formula
_{a}^{b}{\frac {\partial }{\partial \alpha }}f(x,\alpha )\,dx.} By the Heine–Cantor theorem it is uniformly continuous in that set. In other words, for any ε
Leibniz_integral_rule
Monotone maps have countable discontinuities
I {\displaystyle I} that is not closed and bounded (and hence by Heine–Borel theorem not compact). Then the interval can be written as a countable union
Discontinuities of monotone functions
Discontinuities_of_monotone_functions
German polymath and scholar (1777–1855)
Gauss produced the second and third complete proofs of the fundamental theorem of algebra. He also introduced the triple bar symbol (≡) for congruence
Carl_Friedrich_Gauss
Way to divide polygon into smaller parts
mathematical analysis such as for the Bolzano–Weierstrass theorem and Heine–Borel theorem. A finite subdivision rule R {\displaystyle R} consists of
Finite_subdivision_rule
Basic framework of mathematics
generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical
Foundations_of_mathematics
Method of deriving conclusions
inferential steps and often use various rules of inference to establish the theorem they intend to demonstrate. Rules of inference are definitory rules—rules
Rule_of_inference
Number that is not a ratio of integers
Eduard Heine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but
Irrational_number
Formal languages for expressing mathematical theories
is still in active use, was influenced by Automath. QED manifesto Morten Heine Sørensen, Paweł Urzyczyn, Lectures on the Curry–Howard isomorphism, Elsevier
Automath
Mathematical function with no sudden changes
nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but
Continuous_function
Dutch mathematician (1856–1894)
Chebyshev–Markov–Stieltjes inequalities Heine–Stieltjes polynomials Laplace–Stieltjes transform Lebesgue–Stieltjes integral Montel's theorem Riemann–Stieltjes integral
Thomas_Joannes_Stieltjes
relations holding in mathematics. Binet-cauchy identity Binomial inverse theorem Binomial identity Brahmagupta–Fibonacci two-square identity Candido's identity
List of mathematical identities
List_of_mathematical_identities
Technique in mathematical logic
from Glivenko's theorem, proved by Valery Glivenko in 1929. It maps each classical formula φ to its double negation ¬¬φ. Glivenko's theorem states: If φ
Double-negation_translation
Uniform restraint of the change in functions
metric to the integers endowed with the usual Euclidean metric. The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly
Uniform_continuity
and eliminate unnecessary reasoning steps. The associated normalization theorem establishes that every derivation in natural deduction can be transformed
Normal form (natural deduction)
Normal_form_(natural_deduction)
Pathological topological space
another curve. This space is closed and bounded and so compact by the Heine–Borel theorem, but has similar properties to the topologist's sine curve—it too
Topologist's_sine_curve
applied to trigonometry. There is evidence of an early form of Rolle's theorem in his work, though it was stated without a modern formal proof. In his
History_of_calculus
Point to which functions converge in analysis
the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting: lim x → a f ( x ) = L {\displaystyle \lim _{x\to a}f(x)=L}
Limit_of_a_function
Topological space where every sequence has a convergent subsequence
then it is sequentially compact if and only if it is compact (cf. Heine–Borel theorem § Generalization). Here is how to see this, using only the countable
Sequentially_compact_space
Normed vector space that is complete
no infinite–dimensional normed space can be locally compact or have the Heine–Borel property. If x 0 {\displaystyle x_{0}} is a vector and s ≠ 0 {\displaystyle
Banach_space
concept of spectrum in topology, by Elon Lages Lima in 1958 Peixoto's theorem, by Maurício Peixoto in 1959 Costa's minimal surface by Celso José da Costa
List of Brazilian inventions and discoveries
List_of_Brazilian_inventions_and_discoveries
French mathematician (1871–1956)
paradox Borel–Cantelli lemma Borel–Carathéodory theorem Heine–Borel theorem Borel determinacy theorem Borel right process Borel set Borel summation Borel
Émile_Borel
System of mathematical set theory
Heine-Borel theorem and other theorems of analysis. This theorem is Gödel's theorem M4. He proved it by first proving M1, a class existence theorem that
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Type of group and algebra representation
ISBN 978-0-07-084011-9. V. Heine (2007). Group theory in quantum mechanics: an introduction to its present usage. Dover. ISBN 978-0-07-084011-9. V. Heine (1993). Group
Irreducible_representation
Type of topological space in mathematics
particular the real line R) are locally compact as a consequence of the Heine–Borel theorem. Topological manifolds share the local properties of Euclidean spaces
Locally_compact_space
Concept of perpetual recurrence of time
by Friedrich Albert Lange. He was also an admirer of the author Heinrich Heine, one of whose books contains a passage discussing the theory of eternal
Eternal_return
Mathematical-logic system
ISBN 0954300653 Monographs/textbooks for graduate students Sørensen, Morten Heine and Urzyczyn, Paweł (2006), Lectures on the Curry–Howard isomorphism, Elsevier
Lambda_calculus
Barrelled space where closed and bounded subsets are compact
Bornological space – Space where bounded operators are continuous Heine–Borel theorem – Subset of Euclidean space is compact if and only if it is closed
Montel_space
Type of topological space
and compact directly from the definition of compactness, without using Heine-Borel. Every dyadic space (a compact space which is a continuous image of
Polyadic_space
Mathematical inequality relating inner products and norms
existence of u {\displaystyle \mathbf {u} } is guaranteed by the Heine-Borel theorem. If A u = 0 {\displaystyle A\mathbf {u} =\mathbf {0} } then u {\displaystyle
Cauchy–Schwarz_inequality
Used to count, measure, and label
Augustin-Louis Cauchy, Charles Méray (1869), Karl Weierstrass (1872), Eduard Heine (1872), Georg Cantor (1883), and Richard Dedekind (1872). A transcendental
Number
Concept in combinatorics (part of mathematics)
}{\frac {x^{n}}{(q;q)_{n}}},} which are both special cases of the q-binomial theorem: ( a x ; q ) ∞ ( x ; q ) ∞ = ∑ n = 0 ∞ ( a ; q ) n ( q ; q ) n x n . {\displaystyle
Q-Pochhammer_symbol
Relationship between programs and proofs
the type of values returned by a function) is analogous to a logical theorem, subject to hypotheses corresponding to the types of the argument values
Curry–Howard_correspondence
Collection of mathematical objects of finite size
if and only if it is closed and bounded. This is also called the Heine-Borel theorem. In topological vector spaces, a different definition for bounded
Bounded_set
Kennelly–Heaviside layer Hebbian learning – Donald Olding Hebb Heine–Borel theorem – Heinrich Eduard Heine and Émile Borel Heinlein's razor – see Hanlon's razor
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Relation among continuous functions
Arzelà–Ascoli theorem states that a subset of C(X) is compact if and only if it is closed, uniformly bounded and equicontinuous. This is analogous to the Heine–Borel
Equicontinuity
Statement supporting a conclusion
Logic. Cambridge University Press. ISBN 978-1-108-41139-4. Sørensen, Morten Heine; Urzyczyn, Pawel (2006). Lectures on the Curry-Howard Isomorphism. Elsevier
Premise
non-compactness are useless for subsets of Euclidean space Rn: by the Heine–Borel theorem, every bounded closed set is compact there, which means that γ(X)
Measure_of_non-compactness
Branch of topology
a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem). Every continuous image of a compact space is compact. A compact
General_topology
Polynomial sequence
+\beta +1)} . The other solution involves the logarithm function. Bochner's theorem states that the Jacobi polynomials are uniquely characterized as polynomial
Jacobi_polynomials
Set of points equidistant from a center
‖x‖, so it is closed; Sn is also bounded, so it is compact by the Heine–Borel theorem. More generally, in a metric space (E,d), the sphere of center x
Sphere
Symbolic logic system
showing which theorems still do hold in minimal logic, often making implicit use of the valid currying rule and the deduction theorem. By implication
Minimal_logic
compact (i.e., every sequence has a convergent subsequence) and the Heine–Borel theorem. Borel 1. A Borel measure is a measure whose domain is the Borel
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Alternative decimal expansion of 1
real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872. The above approach to decimal expansions
0.999...
Property of topological spaces
subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, connected subsets of R n {\displaystyle \mathbb {R} ^{n}} (for n
Locally_connected_space
Topological vector spaces
every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Study of mathematical analysis seen through computability theory
above notion of computable compactness satisfies an analogue of the Heine–Borel theorem. In particular, the unit interval [ 0 , 1 ] {\displaystyle [0,1]}
Computable_analysis
Type of continuity of a complex-valued function
otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem. It does not satisfy a Hölder condition of any order, however. The
Hölder_condition
Objects that generalize functions
(Rudin 1991, Theorem 3.10). (Strichartz 1994, §2.3); (Trèves 1967). See for example (Hörmander 1983, Theorem 6.1.1). See (Hörmander 1983, Theorem 6.1.2). See
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Austrian cyclist (born 1991)
kilometer. It was composed of Dani Christmas, Anna Plichta, Sara Olsson, Vita Heine and Silvia Valsecchi. Twenty kilometers away, they were joined by Kiesenhofer
Anna_Kiesenhofer
Set of points on a line segment with certain topological properties
therefore a complete metric space. Since it is also totally bounded, the Heine–Borel theorem says that it must be compact. For any point in the Cantor set and
Cantor_set
Three-dimensional fractal
The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. It has Lebesgue measure 0. Because it
Menger_sponge
Locally convex topological vector space
-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of X {\displaystyle X} are weakly compact). Theorem—A locally convex space
Reflexive_space
Mathematical concept
in the weak* topology. If X is a normed space, a version of the Heine-Borel theorem holds. In particular, a subset of the continuous dual is weak* compact
Weak_topology
TVS-embedding as well as an isometry onto its range; furthermore, by Goldstine's theorem (proved in 1938), the range of J is a dense subset of the bidual ( X ′
Semi-reflexive_space
Type whose definition depends on a value
Extensional concepts in intensional type theory (PDF) Sørensen, Morten Heine B.; Urzyczyn, Pawel (1998), Lectures on the Curry-Howard Isomorphism, CiteSeerX 10
Dependent_type
Sequence of points that get progressively closer to each other
Bolzano–Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the Heine–Borel
Cauchy_sequence
Method of software development
test-case generation. The tool is developed by the STUPS group at the Heinrich Heine University Düsseldorf. The Rodin Platform is a tool that supports Event-B
B-Method
Geometric figure which approximates the Earth's shape
Berlin Heidelberg. p. 156. ISBN 978-3-642-12124-1. Retrieved 2021-10-24. Heine, George (September 2013). "Euler and the Flattening of the Earth". Math
Earth_ellipsoid
Mathematical term
not noticed by the larger mathematical community until its rediscovery in Heine in the 1860s. Meanwhile, Cauchy's textbook Cours d'Analyse defined continuity
Microcontinuity
Infinite sum
series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and du Bois-Reymond. Among other prominent contributors
Series_(mathematics)
generalization of the hyperplane separation theorem. Heine A topological vector space is said to have the Heine–Borel property if every closed and bounded
Glossary of functional analysis
Glossary_of_functional_analysis
French jurist and mathematician
Publication: 171–289. The material on de Beaune is on p. 187. Goldstine, Herman Heine (1991), Die Streitschriften, Springer, p. 20, ISBN 9783764323486. Cajori
Florimond_de_Beaune
Generalization of a sequence of points
can be seen as a generalization of the Bolzano–Weierstrass theorem and Heine–Borel theorem. The set of cluster points of a net is equal to the set of
Net_(mathematics)
Sequence of differential equation solutions
special case of Perron's formula with M = 1 {\displaystyle M=1} . The Mehler–Heine formula states: lim n → ∞ n − α L n ( α ) ( z 2 4 n ) = ( z 2 ) − α J α
Laguerre_polynomials
HEINE THEOREM
HEINE THEOREM
Boy/Male
Australian, Danish, Swedish
Home Ruler
Girl/Female
French
Queen.
Female
Yiddish
 Variant spelling of Yiddish Rayna, REINE means "pure." Compare with another form of Reine.
Female
Yiddish
Variant spelling of Yiddish Sheina, SHEINE means "beautiful."
Surname or Lastname
English
English : variant spelling of Hain 1–3.Irish : variant of Hines.Dutch and German : variant of Hein.
Surname or Lastname
English (southwestern)
English (southwestern) : occupational name for a servant, from Middle English hine ‘lad’, ‘servant’ (originally a collective term for a body of servants, from an Old English plural noun, hīwan ‘household’).Americanized spelling of German Hein.
Male
German
Pet form of Old High German Heinrich, HEINO means "home-ruler."
Female
French
 French form of Latin Regina, REINE means "queen." Compare with another form of Reine.
Male
German
Pet form of Old High German Heinrich, HEINER means "home-ruler."
Boy/Male
Teutonic
Rules an estate.
Boy/Male
English
From the bank.
Surname or Lastname
English
English : variant of Hearn 2 and 4.
Boy/Male
Teutonic
Dwells in the hedged enclosure.
Female
Yiddish
Variant spelling of Yiddish Heneh, HENE means "favor; grace."
Girl/Female
Teutonic
Ruler of the home.
Surname or Lastname
English and German
English and German : variant spelling of Hain.
Male
German
Pet form of Old High German Heinrich, HEINZ means "home-ruler."
Boy/Male
Danish, Finnish, German, Swedish
Active; Sprightly
Male
German
Pet form of Old High German Heinrich, HEINE means "home-ruler."
Male
German
Frisian unisex pet form of German Heinrike and Heinrich, HEIKE means "home-ruler."
HEINE THEOREM
HEINE THEOREM
HEINE THEOREM
HEINE THEOREM
HEINE THEOREM
HEINE THEOREM
HEINE THEOREM