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{{Short description|Linear map from a vector space to its field of scalars}}

In [[mathematics]], a '''linear form''' (also known as a '''linear functional''',<ref>{{Harvard citation text|Axler|2015}} p. 101, §3.92</ref> a '''one-form''', or a '''covector''') is a [[linear map]]<ref group=nb>In some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars</ref> from a [[vector space]] to its [[field (mathematics)|field]] of [[scalar (mathematics)|scalars]] (often, the [[real number]]s or the [[complex number]]s).

If {{mvar|V}} is a vector space over a field {{mvar|k}}, the set of all linear functionals from {{mvar|V}} to {{mvar|k}} is itself a vector space over {{mvar|k}} with addition and scalar multiplication defined [[pointwise]]. This space is called the [[dual space]] of {{mvar|V}}, or sometimes the '''algebraic dual space''', when a [[topological dual space]] is also considered. It is often denoted {{math|Hom(''V'', ''k'')}},<ref name=":0">{{Harvard citation text|Tu|2011}} p. 19, §3.1</ref> or, when the field {{mvar|k}} is understood, <math>V^*</math>;<ref>{{Harvard citation text|Katznelson|Katznelson|2008}} p. 37, §2.1.3</ref> other notations are also used, such as <math>V'</math>,<ref>{{Harvard citation text|Axler|2015}} p. 101, §3.94</ref><ref>{{Harvtxt|Halmos|1974}} p. 20, §13</ref> <math>V^{\#}</math> or <math>V^{\vee}.</math><ref name=":0" /> When vectors are represented by [[column vector]]s (as is common when a [[basis (linear algebra)|basis]] is fixed), then linear functionals are represented as [[row vector]]s, and their values on specific vectors are given by [[matrix product]]s (with the row vector on the left).

== Examples ==

The constant [[zero function]], mapping every vector to zero, is trivially a linear functional. Every other linear functional (such as the ones below) is [[Surjective function|surjective]] (that is, its range is all of {{mvar|k}}).
* Indexing into a vector: The second element of a three-vector is given by the one-form <math>[0, 1, 0].</math> That is, the second element of <math>[x, y, z]</math> is <math display=block>[0, 1, 0] \cdot [x, y, z] = y.</math>
* [[Mean]]: The mean element of an <math>n</math>-vector is given by the one-form <math>\left[1/n, 1/n, \ldots, 1/n\right].</math> That is, <math display=block>\operatorname{mean}(v) = \left[1/n, 1/n, \ldots, 1/n\right] \cdot v.</math>
* [[Sampling (signal processing)|Sampling]]: Sampling with a [[Kernel (image processing)|kernel]] can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.
* [[Net present value]] of a net [[cash flow]], <math>R(t),</math> is given by the one-form <math>w(t) = (1 + i)^{-t}</math> where <math>i</math> is the [[Discount window|discount rate]]. That is, <math display=block>\mathrm{NPV}(R(t)) = \langle w, R\rangle = \int_{t=0}^\infty \frac{R(t)}{(1+i)^{t}}\,dt.</math>

=== Linear functionals in R<sup>''n''</sup> ===
Suppose that vectors in the real coordinate space <math>\R^n</math> are represented as column vectors
<math display=block>\mathbf{x} = \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math>

For each row vector <math>\mathbf{a} = \begin{bmatrix}a_1 & \cdots & a_n\end{bmatrix}</math> there is a linear functional <math>f_{\mathbf{a}}</math> defined by
<math display=block>f_{\mathbf{a}}(\mathbf{x}) = a_1 x_1 + \cdots + a_n x_n,</math>
and each linear functional can be expressed in this form.

This can be interpreted as either the matrix product or the dot product of the row vector <math>\mathbf{a}</math> and the column vector <math>\mathbf{x}</math>:
<math display=block>f_{\mathbf{a}}(\mathbf{x}) = \mathbf{a} \cdot \mathbf{x} = \begin{bmatrix}a_1 & \cdots & a_n\end{bmatrix} \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math>

=== Trace of a square matrix ===
The [[Trace (linear algebra)|trace]] <math>\operatorname{tr} (A)</math> of a square matrix <math>A</math> 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 <math>n \times n</math> matrices. The trace is a linear functional on this space because <math>\operatorname{tr} (s A) = s \operatorname{tr} (A)</math> and <math>\operatorname{tr} (A + B) = \operatorname{tr} (A) + \operatorname{tr} (B)</math> for all scalars <math>s</math> and all <math>n \times n</math> matrices <math>A \text{ and } B.</math>

=== (Definite) Integration ===
Linear functionals first appeared in [[functional analysis]], the study of [[Function space|vector spaces of functions]]. A typical example of a linear functional is [[Integral|integration]]: the linear transformation defined by the [[Riemann integral]]
<math display=block>I(f) = \int_a^b f(x)\, dx</math>
is a linear functional from the vector space <math>C[a, b]</math> of continuous functions on the interval <math>[a, b]</math> to the real numbers. The linearity of <math>I</math> follows from the standard facts about the integral:
<math display=block>\begin{align}
     I(f + g) &= \int_a^b[f(x) + g(x)]\, dx = \int_a^b f(x)\, dx + \int_a^b g(x)\, dx = I(f) + I(g) \\
  I(\alpha f) &= \int_a^b \alpha f(x)\, dx = \alpha\int_a^b f(x)\, dx = \alpha I(f).
\end{align}</math>

=== Evaluation ===
Let <math>P_n</math> denote the vector space of real-valued polynomial functions of degree <math>\leq n</math> defined on an interval <math>[a, b].</math> If <math>c \in [a, b],</math> then let <math>\operatorname{ev}_c : P_n \to \R</math> be the '''evaluation functional'''
<math display=block>\operatorname{ev}_c f = f(c).</math>
The mapping <math>f \mapsto f(c)</math> is linear since
<math display=block>\begin{align}
     (f + g)(c) &= f(c) + g(c) \\
  (\alpha f)(c) &= \alpha f(c).
\end{align}</math>

If <math>x_0, \ldots, x_n</math> are <math>n + 1</math> distinct points in <math>[a, b],</math> then the evaluation functionals <math>\operatorname{ev}_{x_i},</math> <math>i = 0, \ldots, n</math> form a [[Basis of a vector space|basis]] of the dual space of <math>P_n</math> ({{harvtxt|Lax|1996}} proves this last fact using [[Lagrange interpolation]]).

=== Non-example ===
A function <math>f</math> having the [[equation of a line]] <math>f(x) = a + r x</math> with <math>a \neq 0</math> (for example, <math>f(x) = 1 + 2 x</math>) is {{em|not}} a linear functional on <math>\R</math>, since it is not [[Linear function|linear]].<ref group="nb">For instance, <math>f(1 + 1) = a + 2 r \neq 2 a + 2 r = f(1) + f(1).</math></ref> It is, however, [[Affine-linear function|affine-linear]].

== Visualization ==
[[File:Gradient 1-form.svg|thumb|200px|Geometric interpretation of a 1-form '''α''' as a stack of [[hyperplane]]s 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 {{color box|purple}} zero plane is through the origin.]]

In finite dimensions, a linear functional can be visualized in terms of its [[level set]]s, 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 [[hyperplane]]s. This method of visualizing linear functionals is sometimes introduced in [[general relativity]] texts, such as [[Gravitation (book)|''Gravitation'']] by {{harvtxt|Misner|Thorne|Wheeler|1973}}.

== Applications ==

=== Application to quadrature ===
If <math>x_0, \ldots, x_n</math> are <math>n + 1</math> distinct points in {{closed-closed|''a'', ''b''}}, then the linear functionals <math>\operatorname{ev}_{x_i} : f \mapsto f\left(x_i\right)</math> defined above form a [[Basis of a vector space|basis]] of the dual space of {{math|''P<sub>n</sub>''}}, the space of polynomials of degree <math>\leq n.</math>  The integration functional {{math|''I''}} is also a linear functional on {{math|''P<sub>n</sub>''}}, and so can be expressed as a linear combination of these basis elements.  In symbols, there are coefficients <math>a_0, \ldots, a_n</math> for which
<math display="block">I(f) = a_0 f(x_0) + a_1 f(x_1) + \dots + a_n f(x_n)</math>
for all <math>f \in P_n.</math> This forms the foundation of the theory of [[numerical quadrature]].<ref>{{harvnb|Lax|1996}}</ref>

=== In quantum mechanics ===
Linear functionals are particularly important in [[quantum mechanics]]. Quantum mechanical systems are represented by [[Hilbert space]]s, which are [[Antilinear|anti]]–[[Linear isomorphism|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]].

=== Distributions ===
In the theory of [[generalized function]]s, certain kinds of generalized functions called [[distribution (mathematics)|distributions]] can be realized as linear functionals on spaces of [[test function]]s.

==Dual vectors and bilinear forms==
[[File:1-form linear functional.svg|thumb|400px|Linear functionals (1-forms) '''α''', '''β''' and their sum '''σ''' and vectors '''u''', '''v''', '''w''', in [[three-dimensional space|3d]] [[Euclidean space]]. The number of (1-form) [[hyperplane]]s intersected by a vector equals the [[inner product]].<ref>{{Harvard citation text|Misner|Thorne|Wheeler|1973}} p. 57</ref>]]

Every non-degenerate [[bilinear form]] on a finite-dimensional vector space {{mvar|V}} induces an [[isomorphism]] {{math|''V'' → ''V''{{i sup|∗}} : ''v'' ↦ ''v''{{sup|∗}}}} such that 
<math display="block"> v^*(w) := \langle v,  w\rangle \quad \forall w \in V ,</math>

where the bilinear form on {{mvar|V}} is denoted <math>\langle \,\cdot\, , \,\cdot\, \rangle</math> (for instance, in [[Euclidean space]], <math>\langle v, w \rangle = v \cdot w</math> is the [[dot product]] of {{mvar|v}} and {{mvar|w}}).

The inverse isomorphism is {{nowrap|''V''{{i sup|∗}} → ''V'' : ''v''{{sup|∗}} ↦ ''v''}}, where {{mvar|v}} is the unique element of {{mvar|V}} such that
<math display="block"> \langle v, w\rangle = v^*(w)</math>
for all <math>w \in V.</math>

The above defined vector {{nowrap|''v''{{sup|∗}} ∈ ''V''{{i sup|∗}}}} is said to be the '''dual vector''' of <math>v \in V.</math>

In an infinite dimensional [[Hilbert space]], analogous results hold by the [[Riesz representation theorem]]. There is a mapping {{nowrap|''V'' ↦ ''V''{{i sup|∗}}}} from {{mvar|V}} into its {{em|[[continuous dual space]]}} ''V''{{i sup|∗}}.

==Relationship to bases==
{{hatnote|Below, we assume that the dimension is finite.  For a discussion of analogous results in infinite dimensions, see [[Schauder basis]].}}

===Basis of the dual space===
Let the vector space {{mvar|V}} have a basis <math>\mathbf{e}_1, \mathbf{e}_2,\dots,\mathbf{e}_n</math>, not necessarily [[orthogonal]]. Then the [[dual space]] <math>V^*</math> has a basis <math>\tilde{\omega}^1,\tilde{\omega}^2,\dots,\tilde{\omega}^n</math> called the [[dual basis]] defined by the special property that
<math display="block"> \tilde{\omega}^i (\mathbf e_j) = \begin{cases}
1 &\text{if}\ i = j\\
0 &\text{if}\ i \neq j.
\end{cases} </math>

Or, more succinctly,
<math display="block"> \tilde{\omega}^i (\mathbf e_j) = \delta_{ij} </math>

where <math>\delta_{ij}</math> is the [[Kronecker delta]]. Here the superscripts of the basis functionals are not exponents but are instead [[Covariance and contravariance of vectors|contravariant]] indices.

A linear functional <math>\tilde{u}</math> belonging to the dual space <math>\tilde{V}</math> can be expressed as a [[linear combination]] of basis functionals, with coefficients ("components") {{math|''u<sub>i</sub>''}},
<math display="block">\tilde{u} = \sum_{i=1}^n u_i \, \tilde{\omega}^i. </math>

Then, applying the functional <math>\tilde{u}</math> to a basis vector <math>\mathbf{e}_j</math> yields
<math display="block">\tilde{u}(\mathbf e_j) = \sum_{i=1}^n \left(u_i \, \tilde{\omega}^i\right) \mathbf e_j = \sum_i u_i \left[\tilde{\omega}^i \left(\mathbf e_j\right)\right] </math>

due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then
<math display="block">\begin{align}
  \tilde{u}({\mathbf e}_j)
&= \sum_i u_i \left[\tilde{\omega}^i \left({\mathbf e}_j\right)\right] \\&
= \sum_i u_i {\delta}_{ij} \\
&= u_j.
\end{align}</math>

So each component of a linear functional can be extracted by applying the functional to the corresponding basis vector.

=== The dual basis and inner product ===
When the space {{mvar|V}} carries an [[inner product]], then it is possible to write explicitly a formula for the dual basis of a given basis. Let {{mvar|V}} have (not necessarily orthogonal) basis <math>\mathbf{e}_1,\dots, \mathbf{e}_n.</math> In three dimensions ({{math|1=''n'' = 3}}), the dual basis can be written explicitly
<math display="block"> \tilde{\omega}^i(\mathbf{v}) = \frac{1}{2} \left\langle \frac { \sum_{j=1}^3\sum_{k=1}^3\varepsilon^{ijk} \, (\mathbf e_j \times \mathbf e_k)} {\mathbf e_1 \cdot \mathbf e_2 \times \mathbf e_3} , \mathbf{v} \right\rangle ,</math>
for <math>i = 1, 2, 3,</math> where ''ε'' is the [[Levi-Civita symbol]] and <math>\langle \cdot , \cdot \rangle</math> the inner product (or [[dot product]]) on {{mvar|V}}.

In higher dimensions, this generalizes as follows
<math display="block"> \tilde{\omega}^i(\mathbf{v}) = \left\langle \frac{\sum_{1 \le i_2 < i_3 < \dots < i_n \le n} \varepsilon^{ii_2\dots i_n}(\star \mathbf{e}_{i_2} \wedge \cdots \wedge \mathbf{e}_{i_n})}{\star(\mathbf{e}_1\wedge\cdots\wedge\mathbf{e}_n)}, \mathbf{v} \right\rangle ,</math>
where <math>\star</math> is the [[Hodge star operator]].

== Over a ring ==
[[Module (mathematics)|Modules]] over a [[Ring (mathematics)|ring]] are generalizations of vector spaces, which removes the restriction that coefficients belong to a [[Field (mathematics)|field]].  Given a module {{mvar|M}} over a ring {{mvar|R}}, a linear form on {{mvar|M}} is a linear map from {{mvar|M}} to {{mvar|R}}, where the latter is considered as a module over itself. The space of linear forms is always denoted {{math|Hom<sub>''k''</sub>(''V'', ''k'')}}, whether {{mvar|k}} is a field or not. It is a [[right module]] if {{mvar|V}} is a left module.

The existence of "enough" linear forms on a module is equivalent to [[Projective module|projectivity]].<ref>{{Cite book|last=Clark|first=Pete L.|url=http://alpha.math.uga.edu/~pete/integral2015.pdf|title=Commutative Algebra|publisher=Unpublished|at=Lemma 3.12}}</ref>  
{{math theorem|math_statement=An {{mvar|R}}-[[Module (mathematics)|module]] {{mvar|M}} is [[projective module|projective]] if and only if there exists a subset <math>A\subset M</math> and linear forms <math>\{f_a\mid a \in A\}</math> such that, for every <math>x\in M,</math> only finitely many <math>f_a(x)</math> are nonzero, and
<math display="block">x=\sum_{a\in A}{f_a(x)a}</math>
|name=Dual Basis Lemma
}}

== Change of field ==
{{anchor|Real and complex linear functionals|Real and imaginary parts of a linear functional}}
{{See also|Linear complex structure|Complexification}}

Suppose that <math>X</math> is a vector space over <math>\Complex.</math> Restricting scalar multiplication to <math>\R</math> gives rise to a real vector space{{sfn|Rudin|1991|pp=57}} <math>X_{\R}</math> called the {{em|[[realification]]}} of <math>X.</math> 
Any vector space <math>X</math> over <math>\Complex</math> is also a vector space over <math>\R,</math> endowed with a [[Linear complex structure|complex structure]]; that is, there exists a real [[vector subspace]] <math>X_{\R}</math> such that we can (formally) write <math>X = X_{\R} \oplus X_{\R}i</math> as <math>\R</math>-vector spaces.

=== Real versus complex linear functionals ===

Every linear functional on <math>X</math> is complex-valued while every linear functional on <math>X_{\R}</math> is real-valued. If <math>\dim X \neq 0</math> then a linear functional on either one of <math>X</math> or <math>X_{\R}</math> is non-trivial (meaning not identically <math>0</math>) if and only if it is surjective (because if <math>\varphi(x) \neq 0</math> then for any scalar <math>s,</math> <math>\varphi\left((s/\varphi(x)) x\right) = s</math>), where the [[Image of a function|image]] of a linear functional on <math>X</math> is <math>\C</math> while the image of a linear functional on <math>X_{\R}</math> is <math>\R.</math> 
Consequently, the only function on <math>X</math> that is both a linear functional on <math>X</math> and a linear function on <math>X_{\R}</math> is the trivial  functional; in other words, <math>X^{\#} \cap X_{\R}^{\#} = \{ 0 \},</math> where <math>\,{\cdot}^{\#}</math> denotes the space's [[algebraic dual space]].  
However, every <math>\Complex</math>-linear functional on <math>X</math> is an [[Linear operator|<math>\R</math>-linear {{em|operator}}]] (meaning that it is [[Additive function|additive]] and [[Real homogeneous|homogeneous over <math>\R</math>]]), but unless it is identically <math>0,</math> it is not an <math>\R</math>-linear {{em|functional}} on <math>X</math> because its range (which is <math>\Complex</math>) is 2-dimensional over <math>\R.</math> Conversely, a non-zero <math>\R</math>-linear functional has range too small to be a <math>\Complex</math>-linear functional as well.

=== Real and imaginary parts ===

If <math>\varphi \in X^{\#}</math> then denote its [[real part]] by <math>\varphi_{\R} := \operatorname{Re} \varphi</math> and its [[imaginary part]] by <math>\varphi_i := \operatorname{Im} \varphi.</math>
Then <math>\varphi_{\R} : X \to \R</math> and <math>\varphi_i : X \to \R</math> are linear functionals on <math>X_{\R}</math> and <math>\varphi = \varphi_{\R} + i \varphi_i.</math> 
The fact that <math>z = \operatorname{Re} z - i \operatorname{Re} (i z) = \operatorname{Im} (i z) + i \operatorname{Im} z</math> for all <math>z \in \Complex</math> implies that for all <math>x \in X,</math>{{sfn|Rudin|1991|pp=57}} 
<math display=block>\begin{alignat}{4}\varphi(x) &= \varphi_{\R}(x) - i \varphi_{\R}(i x) \\
&= \varphi_i(i x) + i \varphi_i(x)\\
\end{alignat}</math>
and consequently, that <math>\varphi_i(x) = - \varphi_{\R}(i x)</math> and <math>\varphi_{\R}(x) = \varphi_i(ix).</math>{{sfn|Narici|Beckenstein|2011|pp=9-11}}

The assignment <math>\varphi \mapsto \varphi_{\R}</math> defines a [[Bijection|bijective]]{{sfn|Narici|Beckenstein|2011|pp=9-11}} <math>\R</math>-linear operator <math>X^{\#} \to X_{\R}^{\#}</math> whose inverse is the map <math>L_{\bull} : X_{\R}^{\#} \to X^{\#}</math> defined by the assignment <math>g \mapsto L_g</math> that sends <math>g : X_{\R} \to \R</math> to the linear functional <math>L_g : X \to \Complex</math> defined by
<math display=block>L_g(x) := g(x) - i g(ix) \quad \text{ for all } x \in X.</math>
The real part of <math>L_g</math> is <math>g</math> and the bijection <math>L_{\bull} : X_{\R}^{\#} \to X^{\#}</math> is an <math>\R</math>-linear operator, meaning that <math>L_{g+h} = L_g + L_h</math> and <math>L_{rg} = r L_g</math> for all <math>r \in \R</math> and <math>g, h \in X_\R^{\#}.</math>{{sfn|Narici|Beckenstein|2011|pp=9-11}} 
Similarly for the imaginary part, the assignment <math>\varphi \mapsto \varphi_i</math> induces an <math>\R</math>-linear bijection <math>X^{\#} \to X_{\R}^{\#}</math> whose inverse is the map <math>X_{\R}^{\#} \to X^{\#}</math> defined by sending <math>I \in X_{\R}^{\#}</math> to the linear functional on <math>X</math> defined by <math>x \mapsto I(i x) + i I(x).</math>

This relationship was discovered by [[Henry Löwig]] in 1934 (although it is usually credited to F. Murray),{{sfn|Narici|Beckenstein|2011|pp=10-11}} and can be generalized to arbitrary [[Finite field extension|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 <math>\varphi : X \to \Complex</math> is a linear functional on <math>X</math> with real part <math>\varphi_{\R} := \operatorname{Re} \varphi</math> and imaginary part <math>\varphi_i := \operatorname{Im} \varphi.</math>

Then <math>\varphi = 0</math> if and only if <math>\varphi_{\R} = 0</math> if and only if <math>\varphi_i = 0.</math>

Assume that <math>X</math> is a [[topological vector space]]. Then <math>\varphi</math> is continuous if and only if its real part <math>\varphi_{\R}</math> is continuous, if and only if <math>\varphi</math>'s imaginary part <math>\varphi_i</math> is continuous. That is, either all three of <math>\varphi, \varphi_{\R},</math> and <math>\varphi_i</math> are continuous or none are continuous. This remains true if the word "continuous" is replaced with the word "[[Bounded linear functional|bounded]]". In particular, <math>\varphi \in X^{\prime}</math> if and only if <math>\varphi_{\R} \in X_{\R}^{\prime}</math> where the prime denotes the space's [[continuous dual space]].{{sfn|Rudin|1991|pp=57}}

Let <math>B \subseteq X.</math> If <math>u B \subseteq B</math> for all scalars <math>u \in \Complex</math> of [[unit length]] (meaning <math>|u| = 1</math>) then<ref group=proof>It is true if <math>B = \varnothing</math> so assume otherwise. Since <math>\left|\operatorname{Re} z\right| \leq |z|</math> for all scalars <math>z \in \Complex,</math> it follows that <math display=inline>\sup_{x \in B} \left|\varphi_{\R}(x)\right| \leq \sup_{x \in B} |\varphi(x)|.</math> If <math>b \in B</math> then let <math>r_b \geq 0</math> and <math>u_b \in \Complex</math> be such that <math>\left|u_b\right| = 1</math> and <math>\varphi(b) = r_b u_b,</math> where if <math>r_b = 0</math> then take <math>u_b := 1.</math>Then <math>|\varphi(b)| = r_b</math> and because <math display=inline>\varphi\left(\frac{1}{u_b} b\right) = r_b</math> is a real number, <math display=inline>\varphi_{\R}\left(\frac{1}{u_b} b\right) = \varphi\left(\frac{1}{u_b} b\right) = r_b.</math> By assumption <math display=inline>\frac{1}{u_b} b \in B</math> so <math display=inline>|\varphi(b)| = r_b \leq \sup_{x \in B} \left|\varphi_{\R}(x)\right|.</math> Since <math>b \in B</math> was arbitrary, it follows that <math display=inline>\sup_{x \in B} |\varphi(x)| \leq \sup_{x \in B} \left|\varphi_{\R}(x)\right|.</math> <math>\blacksquare</math></ref>{{sfn|Narici|Beckenstein|2011|pp=126-128}} <math display=block>\sup_{b \in B} |\varphi(b)| = \sup_{b \in B} \left|\varphi_{\R}(b)\right|.</math> 
Similarly, if <math>\varphi_i := \operatorname{Im} \varphi : X \to \R</math> denotes the complex part of <math>\varphi</math> then <math>i B \subseteq B</math> implies <math display=block>\sup_{b \in B} \left|\varphi_{\R}(b)\right| = \sup_{b \in B} \left|\varphi_i(b)\right|.</math> 
If <math>X</math> is a [[normed space]] with norm <math>\|\cdot\|</math> and if <math>B = \{x \in X : \| x \| \leq 1\}</math> is the closed unit ball then the [[supremum]]s above are the [[operator norm]]s (defined in the usual way) of <math>\varphi, \varphi_{\R},</math> and <math>\varphi_i</math> so that{{sfn|Narici|Beckenstein|2011|pp=126-128}} <math display=block>\|\varphi\| = \left\|\varphi_{\R}\right\| = \left\|\varphi_i \right\|.</math> 
This conclusion extends to the analogous statement for [[Polar set|polars]] of [[balanced set]]s in general [[topological vector space]]s.
* If <math>X</math> is a complex [[Hilbert space]] with a (complex) [[inner product]] <math>\langle \,\cdot\,| \,\cdot\, \rangle</math> that is [[Antilinear map|antilinear]] in its first coordinate (and linear in the second) then <math>X_{\R}</math> becomes a real Hilbert space when endowed with the real part of <math>\langle \,\cdot\,| \,\cdot\, \rangle.</math> Explicitly, this real inner product on <math>X_{\R}</math> is defined by <math>\langle x | y \rangle_{\R} := \operatorname{Re} \langle x | y \rangle</math> for all <math>x, y \in X</math> and it induces the same norm on <math>X</math> as <math>\langle \,\cdot\,| \,\cdot\, \rangle</math> because <math>\sqrt{\langle x | x \rangle_{\R}} = \sqrt{\langle x | x \rangle}</math> for all vectors <math>x.</math> Applying the [[Riesz representation theorem]] to <math>\varphi \in X^{\prime}</math> (resp. to <math>\varphi_{\R} \in X_{\R}^{\prime}</math>) guarantees the existence of a unique vector <math>f_{\varphi} \in X</math> (resp. <math>f_{\varphi_{\R}} \in X_{\R}</math>) such that <math>\varphi(x) = \left\langle f_{\varphi} | \, x \right\rangle</math> (resp. <math>\varphi_{\R}(x) = \left\langle f_{\varphi_{\R}} | \, x \right\rangle_{\R}</math>) for all vectors <math>x.</math> The theorem also guarantees that <math>\left\|f_{\varphi}\right\| = \|\varphi\|_{X^{\prime}}</math> and <math>\left\|f_{\varphi_{\R}}\right\| = \left\|\varphi_{\R}\right\|_{X_{\R}^{\prime}}.</math> It is readily verified that <math>f_{\varphi} = f_{\varphi_{\R}}.</math> Now <math>\left\|f_{\varphi}\right\| = \left\|f_{\varphi_{\R}}\right\|</math> and the previous equalities imply that <math>\|\varphi\|_{X^{\prime}} = \left\|\varphi_{\R}\right\|_{X_{\R}^{\prime}},</math> which is the same conclusion that was reached above.

== In infinite dimensions ==
{{see also|Continuous linear operator}}Below, all [[vector space]]s are over either the [[real number]]s <math>\R</math> or the [[complex number]]s <math>\Complex.</math>

If <math>V</math> is a [[topological vector space]], the space of [[Continuous function|continuous]] linear functionals — the {{em|[[Continuous dual space|continuous dual]]}} — is often simply called the dual space. If <math>V</math> 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 {{em|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.

A linear functional {{mvar|f}} on a (not necessarily [[locally convex]]) [[topological vector space]] {{mvar|X}} is continuous if and only if there exists a continuous seminorm {{mvar|p}} on {{mvar|X}} such that <math>|f| \leq p.</math>{{sfn|Narici|Beckenstein|2011|p=126}}

=== Characterizing closed subspaces ===
Continuous linear functionals have nice properties for [[Real analysis|analysis]]: a linear functional is continuous if and only if its [[Kernel (linear operator)|kernel]] is closed,<ref>{{harvnb|Rudin|1991|loc=Theorem 1.18}}</ref> and a non-trivial continuous linear functional is an [[open map]], even if the (topological) vector space is not complete.{{sfn|Narici|Beckenstein|2011 |p=128}}

==== Hyperplanes and maximal subspaces ====

A vector subspace <math>M</math> of <math>X</math> is called '''maximal''' if <math>M \subsetneq X</math> (meaning <math>M \subseteq X</math> and <math>M \neq X</math>) and does not exist a vector subspace <math>N</math> of <math>X</math> such that <math>M \subsetneq N \subsetneq X.</math> A vector subspace <math>M</math> of <math>X</math> is maximal if and only if it is the kernel of some non-trivial linear functional on <math>X</math> (that is, <math>M = \ker f</math> for some linear functional <math>f</math> on <math>X</math> that is not identically {{math|0}}).  An '''affine hyperplane''' in <math>X</math> is a translate of a maximal vector subspace.  By linearity, a subset <math>H</math> of <math>X</math> is a affine hyperplane if and only if there exists some non-trivial linear functional <math>f</math> on <math>X</math> such that <math>H = f^{-1}(1) = \{ x \in X : f(x) = 1 \}.</math>{{sfn|Narici|Beckenstein|2011|pp=10-11}} 
If <math>f</math> is a linear functional and <math>s \neq 0</math> is a scalar then <math>f^{-1}(s) = s \left(f^{-1}(1)\right) = \left(\frac{1}{s} f\right)^{-1}(1).</math> This equality can be used to relate different level sets of <math>f.</math> Moreover, if <math>f \neq 0</math> then the kernel of <math>f</math> can be reconstructed from the affine hyperplane <math>H := f^{-1}(1)</math> by <math>\ker f = H - H.</math>

==== 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.

{{math theorem|name=Theorem{{sfn|Rudin|1991|pp=63-64}}{{sfn|Narici|Beckenstein|2011|pp=1-18}}|math_statement=
If <math>f, g_1, \ldots, g_n</math> are linear functionals on {{mvar|X}}, then the following are equivalent:

#{{mvar|f}} can be written as a [[linear combination]] of <math>g_1, \ldots, g_n</math>; that is, there exist scalars <math>s_1, \ldots, s_n</math> such that <math>sf = s_1 g_1 + \cdots + s_n g_n</math>;
#<math>\bigcap_{i=1}^{n} \ker g_i \subseteq \ker f</math>;
#there exists a real number {{mvar|r}} such that <math>|f(x)| \leq r g_i (x)</math> for all <math>x \in X</math> and all <math>i = 1, \ldots, n.</math>
}}

If {{mvar|f}} is a non-trivial linear functional on {{mvar|X}} with kernel {{mvar|N}}, <math>x \in X</math> satisfies <math>f(x) = 1,</math> and {{mvar|U}} is a [[Balanced set|balanced]] subset of {{mvar|X}}, then <math>N \cap (x + U) = \varnothing</math> if and only if <math>|f(u)| < 1</math> for all <math>u \in U.</math>{{sfn |Narici|Beckenstein|2011|p=128}}

=== Hahn–Banach theorem ===
{{Main|Hahn–Banach theorem}}

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 <math>\R.</math>  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,

{{math theorem|name=Hahn–Banach dominated extension theorem{{sfn|Narici|Beckenstein|2011|pp=177-220}}{{harv|Rudin|1991|loc=Th. 3.2}}|math_statement=
If <math>p : X \to \R</math> is a [[sublinear function]], and <math>f : M \to \R</math> is a [[linear functional]] on a [[linear subspace]] <math>M \subseteq X</math> which is dominated by {{mvar|p}} on {{mvar|M}}, then there exists a linear extension <math>F : X \to \R</math> of {{mvar|f}} to the whole space {{mvar|X}} that is dominated by {{mvar|p}}, i.e., there exists a linear functional {{mvar|F}} such that
<math display="block">F(m) = f(m)</math>
for all <math>m \in M,</math> and
<math display="block">|F(x)| \leq p(x)</math>
for all <math>x \in X.</math>
}}

=== Equicontinuity of families of linear functionals ===
Let {{mvar|X}} be a [[topological vector space]] (TVS) with [[continuous dual space]] <math>X'.</math>

For any subset {{math|''H''}} of <math>X',</math> the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
# {{math|''H''}} is [[Equicontinuity|equicontinuous]];
# {{math|''H''}} is contained in the [[Polar set|polar]] of some neighborhood of <math>0</math> in {{mvar|X}};
# the [[Polar set|(pre)polar]] of {{math|''H''}} is a neighborhood of <math>0</math> in {{mvar|X}};

If {{math|''H''}} is an equicontinuous subset of <math>X'</math> then the following sets are also equicontinuous: 
the [[Weak-* topology|weak-*]] closure, the [[Balanced set|balanced hull]], the [[convex hull]], and the [[Absolutely convex set|convex balanced hull]].{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
Moreover, [[Alaoglu's theorem]] implies that the weak-* closure of an equicontinuous subset of  <math>X'</math> is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact).{{sfn|Schaefer|Wolff|1999|loc=Corollary 4.3}}{{sfn|Narici|Beckenstein|2011|pp=225-273}}

== See also ==

* {{annotated link|Discontinuous linear map}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Positive linear functional}}
* {{annotated link|Multilinear form}}
* {{annotated link|Topological vector space}}

==Notes==
=== Footnotes ===
{{reflist|group=nb}}

=== Proofs ===
{{reflist|group=proof}}

== References ==
{{reflist}}

== Bibliography ==

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