Editing Hahn–Banach theorem (section)

===For nonlinear functions===
The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem.

{{Math theorem
| name = {{visible anchor|Mazur–Orlicz theorem}}{{sfn|Narici|Beckenstein|2011|pp=177–220}}
| math_statement = Let <math>p : X \to \R</math> be a [[sublinear function]] on a real or complex vector space <math>X,</math> let <math>T</math> be any set, and let <math>R : T \to \R</math> and <math>v : T \to X</math> be any maps. The following statements are equivalent:
# there exists a real-valued linear functional <math>F</math> on <math>X</math> such that <math>F \leq p</math> on <math>X</math> and <math>R \leq F \circ v</math> on <math>T</math>;
# for any finite sequence <math>s_1, \ldots, s_n</math> of <math>n > 0</math> non-negative real numbers, and any sequence <math>t_1, \ldots, t_n \in T</math> of elements of <math>T,</math> <math display=block>\sum_{i=1}^n s_i R\left(t_i\right) \leq p\left(\sum_{i=1}^n s_i v\left(t_i\right)\right).</math>
}}

The following theorem characterizes when {{em|any}} scalar function on <math>X</math> (not necessarily linear) has a continuous linear extension to all of <math>X.</math>

{{Math theorem
| name = Theorem
| note = {{visible anchor|The extension principle}}{{sfn|Edwards|1995|pp=124-125}}
| math_statement = Let <math>f</math> a scalar-valued function on a subset <math>S</math> of a [[topological vector space]] <math>X.</math>  
Then there exists a continuous linear functional <math>F</math> on <math>X</math> extending <math>f</math> if and only if there exists a continuous seminorm <math>p</math> on <math>X</math> such that 
<math display=block>\left|\sum_{i=1}^n a_i f(s_i)\right| \leq p\left(\sum_{i=1}^n a_is_i\right)</math>
for all positive integers <math>n</math> and all finite sequences <math>a_1, \ldots, a_n</math> of scalars and elements <math>s_1, \ldots, s_n</math> of <math>S.</math>
}}