Linear functional

A linear functional is a linear map from a vector space to its scalar field. It assigns each vector a scalar while preserving vector addition and scalar multiplication.

Vector space: A set with vector addition and scalar multiplication satisfying axioms that support linear combinations. Its elements are the inputs a linear functional assigns scalars to.
Coordinate functional: A linear functional that extracts one coordinate relative to a chosen basis. Coordinate functionals provide the standard building blocks for finite-dimensional examples.
Linear programming: An optimization method that maximizes or minimizes a linear objective subject to linear constraints. Its objective is a linear functional evaluated on feasible vectors.
Linear operator: A linear map from a vector space to itself, or to another specified vector space. Unlike a functional, an operator generally returns a vector rather than a scalar.
Linear map: A function between vector spaces that preserves addition and scalar multiplication. A linear functional is the special case whose output space is the scalar field.
Evaluation map: A map that sends a function to its value at a specified input. Evaluation at a point is a linear functional on many spaces of functions.
Lagrange multiplier: A scalar used to enforce a constraint in constrained optimization. Constraint derivatives act as functionals that shape optimality conditions.
Affine functional: A scalar-valued affine map that preserves affine combinations but need not send zero to zero. Its constant offset distinguishes it from a linear functional.
Dual space: The vector space of all linear functionals from a vector space to its scalar field. It collects every possible linear functional on the given space.
Annihilator (linear algebra): The set of linear functionals that vanish on every vector in a specified subspace. It describes which functionals ignore an entire subspace.

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