Linear form

If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise.

Suppose that vectors in the real coordinate space

denote the vector space of real-valued polynomial functions of degree

(Lax (1996) proves this last fact using Lagrange interpolation).

In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplanes.

This method of visualizing linear functionals is sometimes introduced in general relativity texts, such as Gravitation by Misner, Thorne & Wheeler (1973).

[6] Linear functionals are particularly important in quantum mechanics.

A state of a quantum mechanical system can be identified with a linear functional.

Every non-degenerate bilinear form on a finite-dimensional vector space V induces an isomorphism V → V∗ : v ↦ v∗ such that

In an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem.

called the dual basis defined by the special property that

Here the superscripts of the basis functionals are not exponents but are instead contravariant indices.

can be expressed as a linear combination of basis functionals, with coefficients ("components") ui,

In three dimensions (n = 3), the dual basis can be written explicitly

Modules over a ring are generalizations of vector spaces, which removes the restriction that coefficients belong to a field.

The space of linear forms is always denoted Homk(V, k), whether k is a field or not.

The existence of "enough" linear forms on a module is equivalent to projectivity.

[8] Dual Basis Lemma — An R-module M is projective if and only if there exists a subset

endowed with a complex structure; that is, there exists a real vector subspace

This relationship was discovered by Henry Löwig in 1934 (although it is usually credited to F. Murray),[11] and can be generalized to arbitrary finite extensions of a field in the natural way.

is the closed unit ball then the supremums above are the operator norms (defined in the usual way) of

This conclusion extends to the analogous statement for polars of balanced sets in general topological vector spaces.

[13] Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed,[14] and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.

is maximal if and only if it is the kernel of some non-trivial linear functional on

is a affine hyperplane if and only if there exists some non-trivial linear functional

Any two linear functionals with the same kernel are proportional (i.e. scalar multiples of each other).

[15] Any (algebraic) linear functional on a vector subspace can be extended to the whole space; for example, the evaluation functionals described above can be extended to the vector space of polynomials on all of

However, this extension cannot always be done while keeping the linear functional continuous.

of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that

[19] Moreover, Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of