dual space

English

Noun

dual space (plural dual spaces)

(mathematics) The vector space which comprises the set of linear functionals of a given vector space.
(mathematics) The vector space which comprises the set of continuous linear functionals of a given topological vector space.
- 2011, David E. Stewart, Dynamics with Inequalities, Society for Industrial and Applied Mathematics, page 17,
  The dual space of a Banach space ${\\displaystyle X}$ is the vector space of continuous linear functions ${\\displaystyle X\\rightarrow \\mathbb {R} }$ , which are called functionals. Similar notation is used for duality pairing between the Banach space ${\\displaystyle X}$ and its dual space ${\\displaystyle X'}$ : ${\\displaystyle \\left\\langle u,v\\right\\rangle }$ is the result of applying the functional ${\\displaystyle u\\in X'}$ to ${\\displaystyle v\\in X}$ : ${\\displaystyle \\left\\langle u,v\\right\\rangle =u(v)}$ explicitly uses the fact that ${\\displaystyle u}$ is a function ${\\displaystyle X\\rightarrow \\mathbb {R} }$ .

Usage notes

Defined for all vector spaces, the dual space may, for clarity, be called the algebraic dual space. When defined for a topological vector space, the (algebraic) dual space has a subspace that corresponds to continuous linear functionals and is called the continuous dual space or continuous dual — or simply the dual space.

Synonyms

(vector space of linear functionals): algebraic dual space, dual vector space
(vector space of continuous linear functionals): continuous dual space, continuous dual

单词	dual space
释义	dual space English Noun dual space (plural dual spaces) (mathematics) The vector space which comprises the set of linear functionals of a given vector space. (mathematics) The vector space which comprises the set of continuous linear functionals of a given topological vector space. 2011, David E. Stewart, Dynamics with Inequalities, Society for Industrial and Applied Mathematics, page 17, The dual space of a Banach space ${\\displaystyle X}$ is the vector space of continuous linear functions ${\\displaystyle X\\rightarrow \\mathbb {R} }$ , which are called functionals. Similar notation is used for duality pairing between the Banach space ${\\displaystyle X}$ and its dual space ${\\displaystyle X'}$ : ${\\displaystyle \\left\\langle u,v\\right\\rangle }$ is the result of applying the functional ${\\displaystyle u\\in X'}$ to ${\\displaystyle v\\in X}$ : ${\\displaystyle \\left\\langle u,v\\right\\rangle =u(v)}$ explicitly uses the fact that ${\\displaystyle u}$ is a function ${\\displaystyle X\\rightarrow \\mathbb {R} }$ . Usage notes Defined for all vector spaces, the dual space may, for clarity, be called the algebraic dual space. When defined for a topological vector space, the (algebraic) dual space has a subspace that corresponds to continuous linear functionals and is called the continuous dual space or continuous dual — or simply the dual space. Synonyms (vector space of linear functionals): algebraic dual space, dual vector space (vector space of continuous linear functionals): continuous dual space, continuous dual Further reading Linear form on Wikipedia.Wikipedia
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dual space

English

Noun

Usage notes

Synonyms

Further reading