Editing Linear form (section)

==== 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}}