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Concept in functional analysis
called functional analysis, a complemented subspace of a topological vector space X , {\displaystyle X,} is a vector subspace M {\displaystyle M} for which
Complemented_subspace
Concept in linear algebra
fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle V} equipped
Orthogonal_complement
Mathematical concept
complementary dimensions. In particular, any line is complemented to a hyperplane that contains it, then complemented back. All nonzero vectors are the same, in
Symplectic_vector_space
Algebraic element satisfying some of the criteria of an inverse
{\mathcal {C}}(A)} and a complement subspace, and construct G {\displaystyle G} as follows. For y {\displaystyle y} 's in the former subspace, let G {\displaystyle
Generalized_inverse
Algebraic structure formed from a collection of algebraic structures
{\displaystyle N.} A vector subspace is called uncomplemented if it is not a complemented subspace. For example, every vector subspace of a Hausdorff TVS that
Direct_sum
In mathematics, vector subspace
linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when
Linear_subspace
Normed vector space that is complete
null space. The closed linear subspace M {\displaystyle M} of X {\displaystyle X} is said to be a complemented subspace of X {\displaystyle X} if M {\displaystyle
Banach_space
Topics referred to by the same term
(sometimes called an antonym) Complement (group theory) Complementary subspaces Orthogonal complement Schur complement Complement (complexity), relating to
Complement
Linear subspace generated from a vector acted on by a power series of a matrix
algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under
Krylov_subspace
Bound lattice in which every element has a complement
complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra. A complemented lattice
Complemented_lattice
Type of vector space in math
Mathematics, EMS Press. Lindenstrauss, J.; Tzafriri, L. (1971), "On the complemented subspaces problem", Israel Journal of Mathematics, 9 (2): 263–269, doi:10
Hilbert_space
"Small" subset of a topological space
{\displaystyle A} can also be called a meagre subspace of X {\displaystyle X} , meaning a meagre space when given the subspace topology. Importantly, this is not
Meagre_set
Branch of mathematics that studies abstract algebraic structures
end{bmatrix}}^{\mathsf {T}}} fixed by this homomorphism, but the complement subspace maps to [ 0 1 ] ↦ [ a 1 ] , {\displaystyle {\begin{bmatrix}0\\1\
Representation_theory
Completion of the usual space with "points at infinity"
dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1. Equivalently
Projective_space
Smooth manifold with an inner product on each tangent space
normal subgroup of G, fix any complemented subspace W of the Lie algebra of K within the Lie algebra of G. If this subspace is invariant under the linear
Riemannian_manifold
Vectors mapped to 0 by a linear map
mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map L : V → W between two vector
Kernel_(linear_algebra)
Subspace of n-space whose dimension is (n-1)
dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional
Hyperplane
orthogonal complement of its kernel. The orthogonal complement of its kernel is called the initial subspace and its range is called the final subspace. Partial
Partial_isometry
used. □⊥ 1. Orthogonal complement: If W is a linear subspace of an inner product space V, then W⊥ denotes its orthogonal complement, that is, the linear
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Operation in abstract algebra
A\oplus B.} Note that not every closed subspace is complemented; e.g. c 0 {\displaystyle c_{0}} is not complemented in ℓ ∞ . {\displaystyle \ell ^{\infty
Direct_sum_of_modules
Vector space in mathematics
Theorem. A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with symmetric basis. Several interpolation results are
Interpolation_space
Vector space consisting of affine subsets
linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace U {\displaystyle U} is a vector space obtained by "collapsing" U {\displaystyle
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Quadratic form for which there is a non-zero vector on which the form evaluates to zero
space and W is a subspace of V. Then W is called an isotropic subspace of V if some vector in it is isotropic, a totally isotropic subspace if all vectors
Isotropic_quadratic_form
Theorem about the dual of a Hilbert space
surjective linear isomorphism and homeomorphism. See the article on complemented subspaces for more details. The usual notation for plugging an element g {\displaystyle
Riesz_representation_theorem
Subset whose closure is the whole space
B {\displaystyle B} is dense in C {\displaystyle C} (in the respective subspace topology) then A {\displaystyle A} is also dense in C . {\displaystyle
Dense_set
Mathematical property
Objects are Banach spaces, "A part" is interpreted as a subspace or a complemented subspace, "Similar" is interpreted as linearly homeomorphic. Many
Schröder–Bernstein_property
Concept in linear algebra
is an invariant subspace of T {\displaystyle T} whose orthogonal complement W ⊥ {\displaystyle W^{\perp }} is also an invariant subspace of T . {\displaystyle
Reducing_subspace
Mathematical set with some added structure
structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same mathematical structure
Space_(mathematics)
Algebraic structure in linear algebra
if and only if all its coefficients are zero. Linear subspace A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that
Vector_space
assertion is true for the real numbers R {\displaystyle \mathbb {R} } . Complemented subspace – Concept in functional analysis Direct sum – Algebraic structure
Direct sum of topological groups
Direct_sum_of_topological_groups
Subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics
A decoherence-free subspace (DFS) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they
Decoherence-free_subspaces
Mathematical theory
Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or another manifold. It is generalized
Alexander_duality
complemented subspaces of X {\displaystyle X} and Y , {\displaystyle Y,} respectively, then E ⊗ F {\displaystyle E\otimes F} is a complemented vector subspace of
Projective_tensor_product
Israeli mathematician
Scientific career Workplaces Weizmann Institute of Science Thesis Complemented Subspaces of L p {\displaystyle L_{p}} and Universal Spaces (1976) Doctoral
Gideon_Schechtman
Subset with finite complement
for example, for X Y = 0 {\displaystyle XY=0} in the plane. Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology
Cofiniteness
Theorem on extension of bounded linear functionals
{\displaystyle I.} Then Y {\displaystyle Y} is a closed and complemented vector subspace of X . {\displaystyle X.} Proof Since K I {\displaystyle \mathbf
Hahn–Banach_theorem
Idempotent linear transformation from a vector space to itself
by taking the orthogonal complement. For Banach spaces, a one-dimensional subspace always has a closed complementary subspace. This is an immediate consequence
Projection_(linear_algebra)
Various meanings of the terms
or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they
Orthogonality
Space in mathematics and theoretical physics
collinear. The intersections of any Euclidean linear subspace with its orthogonal complement is the {0} subspace. But the definition from the previous subsection
Pseudo-Euclidean_space
Result about when a matrix can be diagonalized
}} , the orthogonal complement of v1. By Hermiticity, K n − 1 {\displaystyle {\mathcal {K}}^{n-1}} is an invariant subspace of A. To see that, consider
Spectral_theorem
Proceedings of the American Mathematical Society, October 1999. "Complemented Subspaces of Products of Banach spaces", Cornell University Library, February
Alex_Chigogidze
Mathematical space
parameterizes the set of all k {\displaystyle k} -dimensional linear subspaces of an n {\displaystyle n} -dimensional vector space V {\displaystyle V}
Grassmannian
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
In geometry, set whose intersection with every line is a single line segment
{\displaystyle \operatorname {rec} A\cap \operatorname {rec} B} is a linear subspace. If A or B is locally compact then A − B is closed. The notion of convexity
Convex_set
Type of geometric transformation
transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point
Blowing_up
Topology in mathematics
is said to be σ-compact if it is the union of countably many compact subspaces, and locally compact if every point has a neighbourhood with compact closure
Stone–Čech_remainder
Certain topology in mathematics
on Z generates the subspace topology on Z, so that the subspace topology will not be an order topology even though it is the subspace topology of a space
Order_topology
On closed convex subsets in Hilbert space
surjective linear isomorphism and homeomorphism. See the article on complemented subspaces for more details. Petersen, Kaare. "The Matrix Cookbook" (PDF).
Hilbert_projection_theorem
Operation in graph theory
[S]_{i}=0} otherwise. A subspace L {\displaystyle L} of K 4 n {\displaystyle K_{4}^{n}} is called a totally isotropic subspace if dim ( L ) = dim (
Local_complementation
Type of linear error-correcting code
the extended binary Golay code G24 consists of a 12-dimensional linear subspace W of the space V = F24 2 of 24-bit words such that any two distinct elements
Binary_Golay_code
Mathematical generalization of boundedness
structure and S {\displaystyle S} be a subset of X . {\displaystyle X.} The subspace bornology A {\displaystyle {\mathcal {A}}} on S {\displaystyle S} is the
Bornology
Topology made of cocountable subsets
{\displaystyle X} has a countable complement. In this case, the cocountable topology is just the discrete topology. Subspace topology: If Y ⊆ X {\displaystyle
Cocountable_topology
Geometric transformation that preserves lines but not angles nor the origin
affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes
Affine_transformation
Russian-American mathematician
Mathematical Surveys, vol. 27, 1972, pp. 1–19 with M. I. Kadets: Complemented subspaces in Banach spaces, Russian Mathematical Surveys, vol. 28, 1973, pp
Boris_Mityagin
Set whose pairs have minima and maxima
bounded lattice for which every element has a complement is called a complemented lattice. A complemented lattice that is also distributive is a Boolean
Lattice_(order)
Concept in geometry
between two subspaces are the same as the non-trivial angles between their orthogonal complements. Non-trivial angles between the subspaces U {\displaystyle
Angles_between_flats
Vector space of infinite sequences
vector subspace TVS-isomorphic to K N {\displaystyle \textstyle \mathbb {K} ^{\mathbb {N} }} . X {\displaystyle X} contains a complemented vector
Sequence_space
Difference between the dimensions of mathematical object and a sub-object
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of
Codimension
more generally to complemented modular lattices, as follows (von Neumann 1998, Part II). His theorem states that if a complemented modular lattice L has
Continuous_geometry
Complement of an open subset
{\displaystyle x} belongs to the closure of A {\displaystyle A} in the topological subspace A ∪ { x } , {\displaystyle A\cup \{x\},} meaning x ∈ cl A ∪ { x } A {\displaystyle
Closed_set
Particular projective representations of the orthogonal or special orthogonal groups
construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this
Spin_representation
Branch of algebraic geometry
sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For further details see Schubert
Schubert_calculus
Point that belongs to the closure of some given subset of a topological space
{\displaystyle X} is a topological subspace of Y {\displaystyle Y} (that is, X {\displaystyle X} is endowed with the subspace topology induced on it by Y {\displaystyle
Adherent_point
Vector spaces associated to a matrix
m\times n} matrix with components from F {\displaystyle F} is a linear subspace of the m-space F m {\displaystyle F^{m}} . The dimension of the column
Row_and_column_spaces
Linear operator defined on a dense linear subspace
operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always)
Unbounded_operator
Reducing subspace Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth
Outline_of_linear_algebra
Basic subset of a topological space
can be given its own topology (called the 'subspace topology') defined by "a set U is open in the subspace topology on Y if and only if U is the intersection
Open_set
Subset which is both open and closed
\mathbb {R} .} The topology on X {\displaystyle X} is inherited as the subspace topology from the ordinary topology on the real line R . {\displaystyle
Clopen_set
Type of topology in mathematics
that their topology is uniquely determined by the family of all finite subspaces. This makes them a generalization of finite topological spaces. Alexandrov-discrete
Alexandrov_topology
is called a complement of a {\displaystyle a} . In contrast to a Boolean algebra, a complemented lattice may have more than one complement for a given
Bounded_lattice
Upper bound on a graph's Shannon capacity
restricting c {\displaystyle c} to the subspace spanned by vectors u i {\displaystyle u_{i}} ; this subspace is at most n {\displaystyle n} -dimensional
Lovász_number
Concept in geometry
projective subspaces are often called affine subspaces of the projective space P, as opposed to the infinite or ideal subspaces, which are the subspaces of the
Hyperplane_at_infinity
Generalization of perpendicularity
{\displaystyle B} . The largest subspace of V {\displaystyle V} that is orthogonal to a given subspace is its orthogonal complement. Given a module M {\displaystyle
Orthogonality_(mathematics)
Vector space with a notion of nearness
(necessarily) Hausdorff TVS. Every vector subspace of X {\displaystyle X} that is an algebraic complement of cl X { 0 } {\displaystyle \operatorname
Topological_vector_space
Examples of topological spaces
not contain p, or A contains all but a finite number of points of X. The subspace X ∖ { p } {\displaystyle X\setminus \{p\}} has the discrete topology and
Fort_space
Mapping from a Euclidean space to itself
a reflection can be described either by the subspace that remains fixed or by its orthogonal complement, whose vectors are reversed. In the preceding
Reflection_(mathematics)
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
Vector satisfying some of the criteria of an eigenvector
independent generalized eigenvectors which form a basis for an invariant subspace of V {\displaystyle V} . Using generalized eigenvectors, a set of linearly
Generalized_eigenvector
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
In mathematics, vector space of linear forms
algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called
Dual_space
Types of mappings in mathematics
zero is a vector subspace of X , {\displaystyle X,} called the null space or kernel of the functional, or the orthogonal complement of x → , {\displaystyle
Functional_(mathematics)
0}{\frac {M(2t)}{M(t)}}<\infty .} We denote by h M {\displaystyle h_{M}} the subspace of scalar sequences ( a n ) n = 1 ∞ ∈ ℓ M {\displaystyle (a_{n})_{n=1}^{\infty
Orlicz_sequence_space
Branch of topology
that generates them. Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets
General_topology
Partition of space by hyperplanes
written L(A), is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are S itself, all the individual
Arrangement_of_hyperplanes
Location of a discrete degeneracy between two electronic states
displacements in a two dimensional subspace of the nuclear coordinate space. The two-dimensional degeneracy lifting subspace is referred to as the branching
Conical_intersection
invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left
Reflexive_operator_algebra
Topological space
spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, complete metric space. (To see this, note that in such
Cantor_space
protection with virtual-address enhancement Set address space control fast Subspace group Called-space identification Checksum Compare and move extended Immediate
IBM Enterprise Systems Architecture
IBM_Enterprise_Systems_Architecture
handle. Subspace weapons are a class of directed energy weapons that directly affect subspace. The weapons can produce actual tears in subspace, and are
Weapons_in_Star_Trek
Algebra based on a vector space with a quadratic form
unital associative algebra with the additional structure of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions
Clifford_algebra
invariant subspace claim. In fact, one can conclude something stronger. The range of EB is actually a reducing subspace of A, i.e. its orthogonal complement is
Quasinormal_operator
Mathematical representation
invariant subspace of H1(Dn) (under the action of Bn) is primitive and infinite cyclic. Let π : H1(Dn) → Z be the projection onto this invariant subspace. Then
Burau_representation
Number of values in the final calculation of a statistic that are free to vary
least-squares projection onto the (n − 1)-dimensional orthogonal complement of this subspace, and has n − 1 degrees of freedom. In statistical testing applications
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Theory of logic to account for observations from quantum theory
correspond to the lattice of closed subspaces of H; the negation of a proposition V is the orthogonal complement V⊥. The space Q of quantum propositions
Quantum_logic
Topological space which is a generalization of certain compact spaces
that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact
Paracompact_space
Concerns the decomposition of representations of a finite group into irreducible pieces
category of representations of G over F is semi-simple. Let U be a subspace of V complement of W. Let p 0 : V → W {\displaystyle p_{0}:V\to W} be the projection
Maschke's_theorem
Representation of a Lie algebra as a set of linear transformations
every invariant subspace of V has an invariant complement. (That is, if W is an invariant subspace, then there is another invariant subspace P such that V
Lie_algebra_representation
Vector space with generalized dot product
{\displaystyle {\overline {H}}.} This means that H {\displaystyle H} is a linear subspace of H ¯ , {\displaystyle {\overline {H}},} the inner product of H {\displaystyle
Inner_product_space
Design method of discrete wavelet transforms
completeness and regularity relations. Self-similarity in time demands that each subspace Vk is invariant under shifts by integer multiples of 2k. That is, for each
Multiresolution_analysis
All points and limit points in a subset of a topological space
if T {\displaystyle T} is a subspace of X {\displaystyle X} (meaning that T {\displaystyle T} is endowed with the subspace topology that X {\displaystyle
Closure_(topology)
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COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
COMPLEMENTED SUBSPACE
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