Editing Linear form (section)

=== 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.