Editing Linear form (section)

==== 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>