Editing Linear form (section)

== 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.