Editing Linear form (section)

=== 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}}