Dual space

The dual space of a vector space is the vector space of all linear functionals from it to its scalar field. Its elements encode linear measurements of vectors.

Vector space: A set with addition and scalar multiplication satisfying axioms that support linear combinations. A dual space is defined from a vector space and inherits its scalar field.
Dual basis: The basis of a dual space whose functionals evaluate to one on a matching basis vector and zero on the others. It turns any basis of the original space into explicit coordinates for its dual.
Lagrange multiplier: A scalar or vector parameter used to impose constraints in an optimization problem. Constraint gradients act as covectors that combine to balance an objective's derivative.
Double dual space: The dual space of a vector space's dual. The canonical map into the double dual is an isomorphism in finite dimensions but not always in infinite dimensions.
Pullback: The operation that transfers functions or covectors backward along a map. A linear map induces a map between dual spaces by composing functionals with that map.
Linear functional: A linear map from a vector space to its scalar field. Each element of the dual space is exactly one linear functional.
Evaluation map: A map that sends a function or functional to its value at a specified input. Applying a dual vector to an original vector is the basic pairing between the spaces.
Differential form: A smoothly varying alternating multilinear function on tangent vectors of a manifold. A one-form is a field of covectors, hence a pointwise element of a cotangent space.
Reflexive space: A normed space whose canonical map onto its double dual is surjective. Reflexivity captures when a space and its double dual coincide in functional analysis.
Rank-nullity theorem: The theorem that a linear map's rank plus its nullity equals the dimension of its domain. Dual maps preserve rank in finite dimensions, linking the constraints they impose to the original map.

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