dual space
In mathematics, the 'dual space' concept is pivotal in understanding the relationship between vectors and linear functionals, with applications in various branches like linear algebra, functional analysis, and differential geometry.
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Definition
C1Linear Algebra
(technical, academic)The set of all linear functionals that map vectors to scalars, forming a vector space itself.
Example
- In linear algebra, every vector space has an associated dual space composed of linear functionals.
C2Functional Analysis
(technical, academic)The set of all continuous linear functionals on a topological vector space.
Example
- The dual space in functional analysis consists of continuous linear mappings that respect the topology of the original space.
C2Differential Geometry
(technical, academic)The cotangent space at a point on a manifold, containing all covectors at that point.
Example
- In differential geometry, the dual space of a tangent space at a point is called the cotangent space.
Similar
Terms that have similar or relatively close meanings to "dual space":
factor spacedual numberhalf spacephase spacefree spaceempty space