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Subset of a topological space whose closure is compact
a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact. Every
Relatively_compact_subspace
Type of mathematical space
space Precompact set - also called totally bounded Quasi-compact morphism Relatively compact subspace Totally bounded Let X = {a, b} ∪ N {\displaystyle \mathbb
Compact_space
Type of topological space in mathematics
These are compact only if they are finite. All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology
Locally_compact_space
Topics referred to by the same term
Precompact set may refer to: Relatively compact subspace, a subset whose closure is compact Totally bounded set, a subset that can be covered by finitely
Precompact_set
Generalization of compactness
complete. Compact space Locally compact space Measure of non-compactness Orthocompact space Paracompact space Relatively compact subspace Sutherland
Totally_bounded_space
Type of continuous linear operator
Y)} is a closed linear subspace of B ( X , Y ) {\displaystyle B(X,Y)} in the operator norm. Equivalently, if a sequence of compact operators T n : X → Y
Compact_operator
Mathematical space with a notion of closeness
Linear subspace – In mathematics, vector subspace Pointless topology Quasitopological space – Function in topology Relatively compact subspace – Subset
Topological_space
Functional analysis concept
T\in L(H)} is said to be a compact operator if the image of each bounded set under T {\displaystyle T} is relatively compact. If X {\displaystyle X} and
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Topological vector spaces
}(U)} is endowed with the subspace topology induced on it by C i ( U ) {\displaystyle C^{i}(U)} . If the family of compact sets K = { U ¯ 1 , U ¯ 2
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Vector space with a notion of nearness
finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it
Topological_vector_space
{\displaystyle K} of C ( X ) {\displaystyle {\mathcal {C}}(X)} is relatively compact if and only if it is bounded in the norm of C ( X ) , {\displaystyle
Space of continuous functions on a compact space
Space_of_continuous_functions_on_a_compact_space
Result about when a matrix can be diagonalized
Hermiticity, K n − 1 {\displaystyle {\mathcal {K}}^{n-1}} is an invariant subspace of A. To see that, consider any k ∈ K n − 1 {\displaystyle k\in {\mathcal
Spectral_theorem
Theory in functional analysis
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert
Spectral theory of compact operators
Spectral_theory_of_compact_operators
On when a family of real, continuous functions has a uniformly convergent subsequence
uniformly on each compact subset of X {\displaystyle X} . Let C c ( X , Y ) {\displaystyle {\mathcal {C}}_{c}(X,Y)} be the subspace of F ( X , Y ) {\displaystyle
Arzelà–Ascoli_theorem
Baire space Banach–Mazur game Meagre set Comeagre set Compact space Relatively compact subspace Heine–Borel theorem Tychonoff's theorem Finite intersection
List of general topology topics
List_of_general_topology_topics
Normed vector space that is complete
{\displaystyle K:={\overline {\operatorname {co} }}S} of this compact subset is compact. The vector subspace X := span S = span { e 1 , e 2 , … } {\displaystyle
Banach_space
Type of vector space in math
subspace of L2(D); in fact, it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact
Hilbert_space
Mathematical generalization of boundedness
topological space X {\displaystyle X} is called relatively compact if its closure is a compact subspace of X . {\displaystyle X.} For any topological space
Bornology
Mathematical space with a notion of distance
number lemma, which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover. Unlike in the
Metric_space
between normed spaces which is not bounded below on any infinite-dimensional subspace. Let X and Y be normed linear spaces, and denote by B(X,Y) the space of
Strictly_singular_operator
Topology where a set is open if it contains a particular point
if X if infinite it is not weakly countably compact. Locally compact but not locally relatively compact. If x ∈ X {\displaystyle x\in X} , then the set
Particular_point_topology
Theorem in functional analysis
of a complete Hausdorff space is compact if (and only if) it is closed and totally bounded. Importantly, the subspace topology that X ′ {\displaystyle
Banach–Alaoglu_theorem
proper if f − 1 ( C ) {\displaystyle f^{-1}(C)} is a compact set in X for any compact subspace C of Y. Proximity space A proximity space (X, d) is a
Glossary_of_general_topology
Function that "converges" to periodicity
relation to a locally compact abelian group G becomes that of a function F in L∞(G), such that its translates by G form a relatively compact set. Equivalently
Almost_periodic_function
Functions that send open (resp. closed) subsets to open (resp. closed) subsets
with the subspace topology induced on it by f {\displaystyle f} 's codomain Y . {\displaystyle Y.} Every strongly open map is a relatively open map.
Open_and_closed_maps
Dual pair of vector spaces
{\displaystyle B} is a vector subspace of X {\displaystyle X} then so too is B ∘ {\displaystyle B^{\circ }} a vector subspace of Y . {\displaystyle Y.} If
Dual_system
Group that is a topological space with continuous group operations
a locally compact commutative group, then for any neighborhood N in G of the identity element, there exists a symmetric relatively compact neighborhood
Topological_group
Type of topological vector space
properties are equivalent: A {\displaystyle A} is equicontinuous; relatively weakly compact; strongly bounded; weakly bounded. The 0-neighborhood bases in
Barrelled_space
Swedish mathematician and concert pianist
The basis problem and the approximation problem and later the invariant subspace problem for Banach spaces. In solving these problems, Enflo developed new
Per_Enflo
Concept in topology
from X to its image in βX is a homeomorphism to an open subspace if and only if X is locally compact Hausdorff. The Stone–Čech construction can be performed
Stone–Čech_compactification
Branch of mathematics that studies abstract algebraic structures
(say) a group G {\displaystyle G} , and W {\displaystyle W} is a linear subspace of V {\displaystyle V} that is preserved by the action of G {\displaystyle
Representation_theory
Generalization of topological interior
is relatively open iff it is equal to its relative interior. Note that when aff ( S ) {\displaystyle \operatorname {aff} (S)} is a closed subspace of
Relative_interior
Relation among continuous functions
of X σ ′ {\displaystyle X_{\sigma }^{\prime }} is a compact metrizable space (under the subspace topology). If in addition X {\displaystyle X} is metrizable
Equicontinuity
Type of continuity of a complex-valued function
\Omega }.} Moreover, this inclusion is compact, meaning that bounded sets in the ‖ · ‖0,β norm are relatively compact in the ‖ · ‖0,α norm. This is a direct
Hölder_condition
Linear map from a vector space to its field of scalars
of X ′ {\displaystyle X'} is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact). Discontinuous linear map Locally convex
Linear_form
Double cover Lie group of the special orthogonal group
the even subspace corresponds to the bosons. The representations of the action of the spin group on the spinor space can be built in a relatively straightforward
Spin_group
Structure in functional analysis
{\displaystyle K:={\overline {\operatorname {co} }}S} of this compact subset is compact. The vector subspace X := span S {\displaystyle X:=\operatorname {span}
Complete topological vector space
Complete_topological_vector_space
Group homomorphism into the general linear group over a vector space
where the relatively weak Zariski topology causes many technical complications. Non-compact topological groups — The class of non-compact groups is too
Group_representation
Product of any collection of compact topological spaces is compact
construction is the Stone–Čech compactification. Conversely, all subspaces of compact Hausdorff spaces are completely regular Hausdorff, so this characterizes
Tychonoff's_theorem
Property of topological spaces
and only if for every open set U, the connected components of U (in the subspace topology) are open. It follows, for instance, that a continuous function
Locally_connected_space
Generalization of boundedness
{\displaystyle H} is equicontinuous. C {\displaystyle C} is a convex compact Hausdorff subspace of X {\displaystyle X} and for every c ∈ C , {\displaystyle c\in
Bounded set (topological vector space)
Bounded_set_(topological_vector_space)
Direct sum of simple Lie algebras
{\mathfrak {g}}} is a compact form and h ⊂ g {\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}} a maximal abelian subspace. One can show (for example
Semisimple_Lie_algebra
Smallest convex set containing a given set
Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The
Convex_hull
Convex hull of a finite set of points in a Euclidean space
in a proper affine subspace of R n {\displaystyle \mathbb {R} ^{n}} and the polytope can be studied as an object in this subspace. In this case, there
Convex_polytope
Description in Riemannian geometry
p ) {\displaystyle K(\sigma _{p})} depends on a two-dimensional linear subspace σ p {\displaystyle \sigma _{p}} of the tangent space at a point p {\displaystyle
Sectional_curvature
Notion in metric geometry
to the metric induced by the ℓ∞ norm. (If d is bounded, then δ is the subspace metric induced by the metric induced by the ℓ∞ norm. If d is not bounded
Tight_span
Characterization of surjectivity
= 0 } {\displaystyle \ker p:=\left\{x\in X:p(x)=0\right\}} is a linear subspace of X {\displaystyle X} . If p {\displaystyle p} is continuous then the
Surjection_of_Fréchet_spaces
Decomposition of periodic functions
291. Oppenheim & Schafer 2010, p. 55. "Characterizations of a linear subspace associated with Fourier series". MathOverflow. 2010-11-19. Retrieved 2014-08-08
Fourier_series
Algebraic structure designed for geometry
defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra. For compactness, we'll use a single capital letter
Geometric_algebra
Book by Lynn Steen
connected sets Gustin's sequence space Roy's lattice space Roy's lattice subspace Cantor's leaky tent Cantor's teepee Pseudo-arc Miller's biconnected set
Counterexamples_in_Topology
Mathematical study of linear operators
1{\text{ is not invertible}}\}.} Invariant subspace Functional calculus Spectral theory Resolvent formalism Compact operator Fredholm theory of integral equations
Operator_theory
Topological vector space in which every closed and bounded subset is complete
quasi-complete locally convex TVS then the set of all compact linear maps of X into Y is a closed vector subspace of L b ( X ; Y ) {\displaystyle L_{b}(X;Y)}
Quasi-complete_space
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
the dual of a nuclear space is a compact metrizable set (for the strong dual topology). Every nuclear space is a subspace of a product of Hilbert spaces
Nuclear_space
Branch of mathematics
infinite-dimensional Banach spaces by examining their finite-dimensional subspaces and quotient spaces. John von Neumann in 1942 studied the asymptotic behavior
Asymptotic_geometry
Function spaces generalizing finite-dimensional p norm spaces
{\displaystyle V\subseteq L^{\infty }(\mu )} is a vector subspace, then V {\displaystyle V} is a closed subspace of L p ( μ ) {\displaystyle L^{p}(\mu )} if and
Lp_space
Mathematical transform that expresses a function of time as a function of frequency
integral Eq.1 does not exist. However, the Fourier transform on the dense subspace L 1 ∩ L 2 ( R ) ⊂ L 2 ( R ) {\displaystyle L^{1}\cap L^{2}(\mathbb {R}
Fourier_transform
Barrelled space where closed and bounded subsets are compact
semi-Montel space or perfect if every bounded subset is relatively compact. A subset of a TVS is compact if and only if it is complete and totally bounded.
Montel_space
Holomorphic functions in infinite dimensions
finite-dimensional subspace, then the series converges uniformly on sufficiently small compact neighborhoods of 0 ∈ Y. However, if the subspace V is permitted
Infinite-dimensional holomorphy
Infinite-dimensional_holomorphy
In mathematics, a partition of a manifold into submanifolds
lamination. One relaxes the condition that the transversals be open, relatively compact subsets of Rq, allowing the transverse coordinates yα to take their
Foliation
Neural network that learns efficient data encoding in an unsupervised manner
layer linear autoencoders have a latent space whose vectors span the same subspace as the eigenvectors found in Principal component analysis. Geoffrey Hinton
Autoencoder
Differentiable function in functional analysis
subspace of C k ( Ω ; Y ) {\displaystyle C^{k}(\Omega ;Y)} consisting of all maps in C k ( Ω ; Y ) {\displaystyle C^{k}(\Omega ;Y)} that have compact
Differentiable vector-valued functions from Euclidean space
Differentiable_vector-valued_functions_from_Euclidean_space
Mathematical structure
a frame is a set of one-dimensional subspaces Li = F·vi such that any k of them generate a k-dimensional subspace. Now an ordered frame L1, ..., Ln defines
Building_(mathematics)
functions Invariant subspace problem – does every bounded operator on a complex Banach space send some non-trivial closed subspace to itself? Kung–Traub
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Topological spaces whose union is a boundary
mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary
Cobordism
linear subspace of Reg([0, T]; X). If X is a Banach space, then the space BV([0, T]; X) of functions of bounded variation forms a dense linear subspace of
Regulated_function
Proposed quantum computer implementation
{\displaystyle \left|\downarrow \uparrow \right\rangle } . The DFS is actually the subspace of two ion states, such that if both ions acquire the same relative phase
Trapped-ion_quantum_computer
Dual space topology of uniform convergence on some sub-collection of bounded subsets
theorem: Every equicontinuous subset of X ′ {\displaystyle X'} is relatively compact for σ ( X ′ , X ) . {\displaystyle \sigma (X',X).} it follows that
Polar_topology
Theory of logic to account for observations from quantum theory
assumed that the orthocomplemented lattice is the lattice of closed linear subspaces of a separable Hilbert space, Constantin Piron, Günther Ludwig and others
Quantum_logic
Branch of mathematics
Lebesgue integral. Other geometrical measures include the curvature and compactness. The concept of length or distance can be generalized, leading to the
Geometry
Type of mathematical functions
then K ^ G {\displaystyle {\hat {K}}_{G}} is the union of K with the relatively compact components of G ∖ K ⊂ G {\displaystyle G\setminus K\subset G} . When
Function of several complex variables
Function_of_several_complex_variables
Function between two metric spaces that only respects their large-scale geometry
map, ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} is quasi-isometric to a subspace of ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} . Two metric spaces M1 and
Quasi-isometry
Class of nonparametric methods
data are sampled. Finding an orthogonal transform onto a low-dimensional subspace B (in the feature space) which minimizes the distributional variance, DICA
Kernel embedding of distributions
Kernel_embedding_of_distributions
Robot used in manufacturing
robot and the 3 orientation coordinates are in the constraint subspace. The motion subspace of lower mobility manipulators may be further decomposed into
Industrial_robot
Unified field theory
symmetry is the symmetry of circular compact dimensions. Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1)
Kaluza–Klein_theory
Kind of linear transformation
{\displaystyle A\in B(X,Y)} the kernel of A {\displaystyle A} is a closed linear subspace of X {\displaystyle X} . If B ( X , Y ) {\displaystyle B(X,Y)} is Banach
Bounded_operator
Property of a mathematical space
universe is localized on a (3 + 1)-dimensional subspace. Thus, the extra dimensions need not be small and compact but may be large extra dimensions. D-branes
Dimension
Algebro-geometric stability condition
Kähler metrics on compact Kähler manifolds, now known as the Calabi conjecture. One formulation of the conjecture is that a compact Kähler manifold X
K-stability
Concept in metric geometry
{asdim} (\mathbb {H} ^{n})=n} . If Y ⊆ X {\displaystyle Y\subseteq X} is a subspace of a metric space X {\displaystyle X} , then asdim ( Y ) ≤ asdim (
Asymptotic_dimension
Type of plane partition
Euclidean case, since the equidistant locus for two points may fail to be subspace of codimension 1, even in the two-dimensional case. A weighted Voronoi
Voronoi_diagram
Group that is also a differentiable manifold with group operations that are smooth
representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the representations
Lie_group
Analytic function on the upper half-plane with a certain behavior under the modular group
forms and its relations for arbitrary Fuchsian groups. New forms are a subspace of modular forms of a fixed level N {\displaystyle N} which cannot be constructed
Modular_form
Important problem in lattice theory
isomorphic to the congruence lattice of some algebra. The lattice Sub V of all subspaces of a vector space V is certainly an algebraic lattice. As the next result
Congruence_lattice_problem
Concept in mathematics
everywhere to a function g {\displaystyle g} taking values in a separable subspace B 0 {\displaystyle B_{0}} of B {\displaystyle B} , and such that the inverse
Bochner_integral
Formulation to quantize gauge field theories in physics
transformations) form a subspace of the space of all (infinitesimal) perturbations; in the non-Abelian case, the embedding of this subspace in the larger space
BRST_quantization
Diode that emits light from an organic compound
the Efficiency of Polariton OLEDs in and Beyond the Single-Excitation Subspace". Advanced Optical Materials. 13 (12) 2403046. arXiv:2404.04257. doi:10
OLED
Single-linkage clustering: a simple agglomerative clustering algorithm SUBCLU: a subspace clustering algorithm WACA clustering algorithm: a local clustering algorithm
List_of_algorithms
Theory of subatomic structure
physicists assume that the observable universe is a four-dimensional subspace of a higher dimensional space. In such models, the force-carrying bosons
String_theory
Topological space that locally resembles Euclidean space
curves and surfaces, including for example all n-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it.
Manifold
Operation in calculus
with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most
Integral
Algebraic structure of set algebra
Mirna; Kunen, Kenneth (1995). "Properties of the class of measure separable compact spaces" (PDF). Fundamenta Mathematicae: 262. If μ {\displaystyle \mu }
Σ-algebra
fact that a high-dimensional state vector belongs to a low-dimensional subspace [1]. Figure below shows the concept of the MOR approximation: finding matrix
Thermal simulations for integrated circuits
Thermal_simulations_for_integrated_circuits
Sum of inverse squares of natural numbers
L2 periodic functions over ( 0 , 1 ) {\displaystyle (0,1)} (i.e., the subspace of square-integrable functions which are also periodic), denoted by { e
Basel_problem
Partial differential equation describing the evolution of temperature in a region
_{mn}} Finally, the sequence {en}n ∈ N spans a dense linear subspace of L2((0, L)). This shows that in effect we have diagonalized the operator
Heat_equation
point is that the vector space V {\displaystyle V} can be decomposed into subspaces V − {\displaystyle V^{-}} (the span of the first n {\displaystyle n} basis
Derivations of the Lorentz transformations
Derivations_of_the_Lorentz_transformations
Axiomatic set theories based on the principles of mathematical constructivism
be a subcountable set. In this theory one may now also quantify over subspaces of spaces like 2 N {\displaystyle 2^{\mathbb {N} }} , which is a third
Constructive_set_theory
Function in discrete mathematics
projection operator method does not produce orthogonal eigenvectors within one subspace. The operator P λ {\displaystyle {\mathcal {P}}_{\lambda }} can be seen
Discrete_Fourier_transform
Ohio State University. OCLC 972879937. Embry, Mary Rodriguez (1964). Subspaces associated with contractions in Hilbert space (PDF). Chapel Hill, NC:
List of African-American mathematicians
List_of_African-American_mathematicians
Tool for digital signal processing
block transform where the length L of basis functions (filters) and the subspace dimension M are the same. Multidimensional filtering, downsampling, and
Filter_bank
Technique in computational quantum field theory
{\displaystyle E} from a proper effective Hamiltonian in P {\displaystyle P} -subspace in favor of eigenvalues of H 0 {\displaystyle H_{0}} . Consequently, the
Light-front computational methods
Light-front_computational_methods
conduit for the donated Future Energy to revive Musashi from a dimensional subspace. She is voiced by Rina Mogami (最上 莉奈, Mogami Rina) in latter two movies
List of Ultraman Cosmos characters
List_of_Ultraman_Cosmos_characters
RELATIVELY COMPACT-SUBSPACE
RELATIVELY COMPACT-SUBSPACE
RELATIVELY COMPACT-SUBSPACE
RELATIVELY COMPACT-SUBSPACE
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RELATIVELY COMPACT-SUBSPACE