Editing Linear form (section)

=== 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.