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. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted Hom(V, k), or, when the field k is understood, ; other notations are also used, such as , or  When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left).
The trace of a square matrix is the sum of all elements on its main diagonal. Matrices can be multiplied by scalars and two matrices of the same dimension can be added together; these operations make a vector space from the set of all matrices. The trace is a linear functional on this space because and for all scalars and all matrices
Geometric interpretation of a 1-form α as a stack of hyperplanes of constant value, each corresponding to those vectors that α maps to a given scalar value shown next to it along with the "sense" of increase. The zero plane is through the origin.
In finite dimensions, a linear functional can be visualized in terms of its level sets, the sets of vectors which map to a given value. 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).
If are distinct points in [a, b], then the linear functionals defined above form a basis of the dual space of Pn, the space of polynomials of degree The integration functional I is also a linear functional on Pn, and so can be expressed as a linear combination of these basis elements. In symbols, there are coefficients for which
Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are anti–isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra–ket notation.
When the space V carries an inner product, then it is possible to write explicitly a formula for the dual basis of a given basis. Let V have (not necessarily orthogonal) basis 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. Given a module M over a ring R, a linear form on M is a linear map from M to R, where the latter is considered as a module over itself. The space of linear forms is always denoted Homk(V, k), whether k is a field or not. It is an right module, if V is a left module.
The existence of "enough" linear forms on a module is equivalent to projectivity.
Dual Basis Lemma — An R-moduleM is projective if and only if there exists a subset and linear forms such that, for every only finitely many are nonzero, and
Suppose that is a vector space over Restricting scalar multiplication to gives rise to a real vector space called the realification of
Any vector space over is also a vector space over endowed with a complex structure; that is, there exists a real vector subspace such that we can (formally) write as -vector spaces.
Real versus complex linear functionals
Every linear functional on is complex-valued while every linear functional on is real-valued. If then a linear functional on either one of or is non-trivial (meaning not identically ) if and only if it is surjective (because if then for any scalar ), where the image of a linear functional on is while the image of a linear functional on is
Consequently, the only function on that is both a linear functional on and a linear function on is the trivial functional; in other words, where denotes the space's algebraic dual space.
However, every -linear functional on is an -linear operator (meaning that it is additive and homogeneous over ), but unless it is identically it is not an -linear functional on because its range (which is ) is 2-dimensional over Conversely, a non-zero -linear functional has range too small to be a -linear functional as well.
Real and imaginary parts
If then denote its real part by and its imaginary part by
Then and are linear functionals on and
The fact that for all implies that for all 
The assignment defines a bijective-linear operator whose inverse is the map defined by the assignment that sends to the linear functional defined by
The real part of is and the bijection is an -linear operator, meaning that and for all and 
Similarly for the imaginary part, the assignment induces an -linear bijection whose inverse is the map defined by sending to the linear functional on defined by
This relationship was discovered by Henry Löwig in 1934 (although it is usually credited to F. Murray), and can be generalized to arbitrary finite extensions of a field in the natural way. It has many important consequences, some of which will now be described.
Properties and relationships
Suppose is a linear functional on with real part and imaginary part
Then if and only if if and only if
Assume that is a topological vector space. Then is continuous if and only if its real part is continuous, if and only if 's imaginary part is continuous. That is, either all three of and are continuous or none are continuous. This remains true if the word "continuous" is replaced with the word "bounded". In particular, if and only if where the prime denotes the space's continuous dual space.
If is a complex Hilbert space with a (complex) inner product that is antilinear in its first coordinate (and linear in the second) then becomes a real Hilbert space when endowed with the real part of Explicitly, this real inner product on is defined by for all and it induces the same norm on as because for all vectors Applying the Riesz representation theorem to (resp. to ) guarantees the existence of a unique vector (resp. ) such that (resp. ) for all vectors The theorem also guarantees that and It is readily verified that Now and the previous equalities imply that which is the same conclusion that was reached above.
If is a topological vector space, the space of continuous linear functionals — the continuous dual — is often simply called the dual space. If is a Banach space, then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual space. In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, but in infinite dimensions the continuous dual is a proper subspace of the algebraic dual.
Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed, and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.
A vector subspace of is called maximal if (meaning and ) and does not exist a vector subspace of such that A vector subspace of is maximal if and only if it is the kernel of some non-trivial linear functional on (that is, for some linear functional on that is not identically 0). An affine hyperplane in is a translate of a maximal vector subspace. By linearity, a subset of is a affine hyperplane if and only if there exists some non-trivial linear functional on such that 
If is a linear functional and is a scalar then This equality can be used to relate different level sets of Moreover, if then the kernel of can be reconstructed from the affine hyperplane by
Relationships between multiple linear functionals
Any two linear functionals with the same kernel are proportional (i.e. scalar multiples of each other).
This fact can be generalized to the following theorem.
Theorem — If are linear functionals on X, then the following are equivalent:
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. The Hahn–Banach family of theorems gives conditions under which this extension can be done. For example,
^It is true if so assume otherwise. Since for all scalars it follows that If then let and be such that and where if then take Then and because is a real number, By assumption so Since was arbitrary, it follows that