Editing Hahn–Banach theorem (section)

===For complex or real vector spaces===

The dominated extension theorem for real linear functionals implies the following alternative statement of the Hahn–Banach theorem that can be applied to linear functionals on real or complex vector spaces. 

{{Math theorem
| name = {{visible anchor|Hahn–Banach theorem for real or complex vector spaces|text=Hahn–Banach theorem}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}<ref>{{harvnb|Rudin|1991}}, Th. 3.2</ref>
| math_statement = Suppose <math>p : X \to \R</math> a [[seminorm]] on a vector space <math>X</math> over the field <math>\mathbf{K},</math> which is either <math>\R</math> or <math>\Complex.</math> 
If <math>f : M \to \mathbf{K}</math> is a linear functional on a vector subspace <math>M</math> such that 
<math display=block>|f(m)| \leq p(m) \quad \text{ for all } m \in M,</math>
then there exists a linear functional <math>F : X \to \mathbf{K}</math> such that
<math display=block>F(m) = f(m) \quad \; \text{ for all } m \in M,</math>
<math display=block>|F(x)| \leq p(x) \quad \;\, \text{ for all } x \in X.</math>
}}

The theorem remains true if the requirements on <math>p</math> are relaxed to require only that for all <math>x, y \in X</math> and all scalars <math>a</math> and <math>b</math> satisfying <math>|a| + |b| \leq 1,</math>{{Sfn|Reed|Simon|1980|p=}}
<math display=block>p(a x + b y) \leq |a| p(x) + |b| p(y).</math>
This condition holds if and only if <math>p</math> is a [[Convex function|convex]] and [[balanced function]] satisfying <math>p(0) \leq 0,</math> or equivalently, if and only if it is convex, satisfies <math>p(0) \leq 0,</math> and <math>p(u x) \leq p(x)</math> for all <math>x \in X</math> and all [[unit length]] scalars <math>u.</math> 

A complex-valued functional <math>F</math> is said to be {{em|{{visible anchor|dominated complex functional|text=dominated by <math>p</math>}}}} if <math>|F(x)| \leq p(x)</math> for all <math>x</math> in the domain of <math>F.</math> 
With this terminology, the above statements of the Hahn–Banach theorem can be restated more succinctly:

:'''Hahn–Banach dominated extension theorem''': If <math>p : X \to \R</math> is a [[seminorm]] defined on a real or complex vector space <math>X,</math> then every [[#dominated complex functional|dominated]] linear functional defined on a vector subspace of <math>X</math> has a dominated linear extension to all of <math>X.</math> In the case where <math>X</math> is a real vector space and <math>p : X \to \R</math> is merely a [[Convex function|convex]] or [[sublinear function]], this conclusion will remain true if both instances of "[[#dominated complex functional|dominated]]" (meaning <math>|F| \leq p</math>) are weakened to instead mean "[[#dominated real functional|dominated {{em|above}}]]" (meaning <math>F \leq p</math>).{{Sfn|Schechter|1996|pp=318-319}}{{Sfn|Reed|Simon|1980|p=}} 

'''Proof'''

The following observations allow the [[#Hahn–Banach dominated extension theorem|Hahn–Banach theorem for real vector spaces]] to be applied to (complex-valued) linear functionals on complex vector spaces.

Every linear functional <math>F : X \to \Complex</math> on a complex vector space is [[Linear form#Real and imaginary parts of a linear functional|completely determined]] by its [[real part]] <math>\; \operatorname{Re} F : X \to \R \;</math> through the formula{{sfn|Narici|Beckenstein|2011|pp=177-183}}<ref group=proof>If <math>z = a + i b \in \Complex</math> has real part <math>\operatorname{Re} z = a</math> then <math>- \operatorname{Re} (i z) = b,</math> which proves that <math>z = \operatorname{Re} z - i \operatorname{Re} (i z).</math> Substituting <math>F(x)</math> in for <math>z</math> and using <math>i F(x) = F(i x)</math> gives <math>F(x) = \operatorname{Re} F(x) - i \operatorname{Re} F(i x).</math> <math>\blacksquare</math></ref> 
<math display=block>F(x) \;=\; \operatorname{Re} F(x) - i \operatorname{Re} F(i x) \qquad \text{ for all } x \in X</math>
and moreover, if <math>\|\cdot\|</math> is a [[Norm (mathematics)|norm]] on <math>X</math> then their [[dual norm]]s are equal: <math>\|F\| = \|\operatorname{Re} F\|.</math>{{sfn|Narici|Beckenstein|2011|pp=126-128}} 
In particular, a linear functional on <math>X</math> extends another one defined on <math>M \subseteq X</math> if and only if their real parts are equal on <math>M</math> (in other words, a linear functional <math>F</math> extends <math>f</math> if and only if <math>\operatorname{Re} F</math> extends <math>\operatorname{Re} f</math>). 
The real part of a linear functional on <math>X</math> is always a {{visible anchor|real-linear functional}} (meaning that it is linear when <math>X</math> is considered as a real vector space) and if <math>R : X \to \R</math> is a real-linear functional on a complex vector space then <math>x \mapsto R(x) - i R(i x)</math> defines the unique linear functional on <math>X</math> whose real part is <math>R.</math> 

If <math>F</math> is a linear functional on a (complex or real) vector space <math>X</math> and if <math>p : X \to \R</math> is a seminorm then{{sfn|Narici|Beckenstein|2011|pp=177-183}}<ref group=proof>Let <math>F</math> be any [[Homogeneous function#Homogeneity|homogeneous]] scalar-valued map on <math>X</math> (such as a linear functional) and let <math>p : X \to \R</math> be any map that satisfies <math>p(u x) = p(x)</math> for all <math>x</math> and [[unit length]] scalars <math>u</math> (such as a seminorm). If <math>|F| \leq p</math> then <math>\operatorname{Re} F \leq |\operatorname{Re} F| \leq |F| \leq p.</math> For the converse, assume <math>\operatorname{Re} F \leq p</math> and fix <math>x \in X.</math> Let <math>r = |F(x)|</math> and pick any <math>\theta \in \R</math> such that <math>F(x) = r e^{i \theta};</math> it remains to show <math>r \leq p(x).</math> Homogeneity of <math>F</math> implies <math>F\left(e^{-i \theta} x\right) = r</math> is real so that <math>\operatorname{Re} F\left(e^{-i \theta} x\right) = F\left(e^{-i \theta} x\right).</math> By assumption, <math>\operatorname{Re} F \leq p</math> and <math>p\left(e^{-i \theta} x\right) = p(x),</math> so that <math>r = \operatorname{Re} F\left(e^{-i \theta} x\right) \leq p\left(e^{-i \theta} x\right) = p(x),</math> as desired. <math>\blacksquare</math></ref> 
<math display=block>|F| \,\leq\, p \quad \text{ if and only if } \quad \operatorname{Re} F \,\leq\, p.</math>
Stated in simpler language, a linear functional is [[#dominated complex functional|dominated]] by a seminorm <math>p</math> if and only if its [[#dominated real functional|real part is dominated above]] by <math>p.</math> 

{{Math proof|title=Proof of [[#Hahn–Banach theorem for real or complex vector spaces|Hahn–Banach for complex vector spaces]] by reduction to real vector spaces{{sfn|Narici|Beckenstein|2011|pp=177-220}}|drop=hidden|proof=
Suppose <math>p : X \to \R</math> is a seminorm on a complex vector space <math>X</math> and let <math>f : M \to \Complex</math> be a linear functional defined on a vector subspace <math>M</math> of <math>X</math> that satisfies <math>|f| \leq p</math> on <math>M.</math> 
Consider <math>X</math> as a real vector space and apply the [[#Hahn–Banach dominated extension theorem|Hahn–Banach theorem for real vector spaces]] to the [[#real-linear functional|real-linear functional]] <math>\; \operatorname{Re} f : M \to \R \;</math> to obtain a real-linear extension <math>R : X \to \R</math> that is also dominated above by <math>p,</math> so that it satisfies <math>R \leq p</math> on <math>X</math> and <math>R = \operatorname{Re} f</math> on <math>M.</math> 
The map <math>F : X \to \Complex</math> defined by <math>F(x) \;=\; R(x) - i R(i x)</math> is a linear functional on <math>X</math> that extends <math>f</math> (because their real parts agree on <math>M</math>) and satisfies <math>|F| \leq p</math> on <math>X</math> (because <math>\operatorname{Re} F \leq p</math> and <math>p</math> is a seminorm). 
<math>\blacksquare</math>
}}

The proof above shows that when <math>p</math> is a seminorm then there is a one-to-one correspondence between dominated linear extensions of <math>f : M \to \Complex</math> and dominated real-linear extensions of <math>\operatorname{Re} f : M \to \R;</math> the proof even gives a formula for explicitly constructing a linear extension of <math>f</math> from any given real-linear extension of its real part.

'''Continuity'''

A linear functional <math>F</math> on a [[topological vector space]]  is [[Continuous linear functional|continuous]] if and only if this is true of its real part <math>\operatorname{Re} F;</math> if the domain is a normed space then <math>\|F\| = \|\operatorname{Re} F\|</math> (where one side is infinite if and only if the other side is infinite).{{sfn|Narici|Beckenstein|2011|pp=126-128}} 
Assume <math>X</math> is a [[topological vector space]] and <math>p : X \to \R</math> is [[sublinear function]]. 
If <math>p</math> is a [[Continuity (topology)|continuous]] sublinear function that dominates a linear functional <math>F</math> then <math>F</math> is necessarily continuous.{{sfn|Narici|Beckenstein|2011|pp=177-183}} Moreover, a linear functional <math>F</math> is continuous if and only if its [[absolute value]] <math>|F|</math> (which is a [[seminorm]] that dominates <math>F</math>) is continuous.{{sfn|Narici|Beckenstein|2011|pp=177-183}} In particular, a linear functional is continuous if and only if it is dominated by some continuous sublinear function.