Dual Space
In mathematics, any vector space, V, has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors. When applied to vector spaces of functions (which typically are infinite-dimensional), dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in the study of functional analysis.
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Some articles on dual space:
... In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ V → V′′ from a normed space V into its continuous double dual V′′, defined by ... Normed spaces for which the map Ψ is a bijection are called reflexive ... When V is a topological vector space, one can still define Ψ(x) by the same formula, for every x ∈ V, however several difficulties arise ...
... Let the vector space V have a basis, not necessarily orthogonal ... Then the dual space V* has a basis called the dual basis defined by the special property that Or, more succinctly, where δ is the Kronecker delta ... A linear functional belonging to the dual space can be expressed as a linear combination of basis functionals, with coefficients ("components") ui, Then, applying the functional to a basis vector ej yields due ...
... or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars ... In general, if V is a vector space over a field k, then a linear functional ƒ is a function from V to k, which is linear for all for all The set of all linear functionals from V to k, Homk(V,k), is itself a ... This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space ...
... The dual space H* is the space of all continuous linear functions from the space H into the base field ... norm, defined by This norm satisfies the parallelogram law, and so the dual space is also an inner product space ... The dual space is also complete, and so it is a Hilbert space in its own right ...
... Let be a convex cone in a real vector space V equipped with an inner product ... The dual cone to C is the set This is also a convex cone ... If C is equal to its dual cone, C is called self-dual ...
Famous quotes containing the words space and/or dual:
“Our passionate preoccupation with the sky, the stars, and a God somewhere in outer space is a homing impulse. We are drawn back to where we came from.”
—Eric Hoffer (1902–1983)
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convulsive,
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—Walt Whitman (1819–1892)