In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (topological) complement in , such that is the direct sum in the category of topological vector spaces. Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional vector spaces.
Every finite-dimensional subspace of a Banach space is complemented, but other subspaces may not. In general, classifying all complemented subspaces is a difficult problem, which has been solved only for some well-known Banach spaces.
The concept of a complemented subspace is analogous to, but distinct from, that of a set complement. The set-theoretic complement of a vector subspace is never a complementary subspace.
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called functional analysis, a complementedsubspace of a topological vector space X , {\displaystyle X,} is a vector subspace M {\displaystyle M} for which...
fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle V} equipped...
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...
{\displaystyle N.} A vector subspace is called uncomplemented if it is not a complementedsubspace. For example, every vector subspace of a Hausdorff TVS that...
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...
complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra. A complemented lattice...
null space. The closed linear subspace M {\displaystyle M} of X {\displaystyle X} is said to be a complementedsubspace of X {\displaystyle X} if M {\displaystyle...
Phonetic complement Complementary, a type of opposite in lexical semantics (sometimes called an antonym) Complement (group theory) Complementary subspaces Orthogonal...
symplectic matrices. Let W be a linear subspace of V. Define the symplectic complement of W to be the subspace W ⊥ = { v ∈ V ∣ ω ( v , w ) = 0 for all ...
Mathematics, EMS Press. Lindenstrauss, J.; Tzafriri, L. (1971), "On the complementedsubspaces problem", Israel Journal of Mathematics, 9 (2): 263–269, doi:10...
{\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...
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...
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...
assertion is true for the real numbers R {\displaystyle \mathbb {R} } Complementedsubspace Direct sum – Operation in abstract algebra composing objects into...
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...
B {\displaystyle B} is dense in C {\displaystyle C} (in the respective subspace topology) then A {\displaystyle A} is also dense in C . {\displaystyle...
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...
end{bmatrix}}^{\mathsf {T}}} fixed by this homomorphism, but the complementsubspace maps to [ 0 1 ] ↦ [ a 1 ] {\displaystyle {\begin{bmatrix}0\\1\end{bmatrix}}\mapsto...
Scientific career Institutions Weizmann Institute of Science Thesis ComplementedSubspaces of L p {\displaystyle L_{p}} and Universal Spaces (1976) Doctoral...
Theorem. A Banach space with unconditional basis is isomorphic to a complementedsubspace of a space with symmetric basis. Several interpolation results are...
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...
vector subspace TVS-isomorphic to K N {\displaystyle \mathbb {K} ^{\mathbb {N} }} . X {\displaystyle X} contains a complemented vector subspace TVS-isomorphic...
or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they...
complemented subspaces of X {\displaystyle X} and Y , {\displaystyle Y,} respectively, then E ⊗ F {\displaystyle E\otimes F} is a complemented vector subspace of...
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