Editing Hahn–Banach theorem (section)

==Applications==

The Hahn–Banach theorem is the first sign of an important philosophy in [[functional analysis]]: to understand a space, one should understand its [[continuous functional]]s.

For example, linear subspaces are characterized by functionals: if {{mvar|X}} is a normed vector space with linear subspace {{mvar|M}} (not necessarily closed) and if <math>z</math> is an element of {{mvar|X}} not in the [[Closure (topology)|closure]] of {{mvar|M}}, then there exists a continuous linear map <math>f : X \to \mathbf{K}</math> with <math>f(m) = 0</math> for all <math>m \in M,</math> <math>f(z) = 1,</math> and <math>\|f\| = \operatorname{dist}(z, M)^{-1}.</math>  (To see this, note that <math>\operatorname{dist}(\cdot, M)</math> is a sublinear function.)  Moreover, if <math>z</math> is an element of {{mvar|X}}, then there exists a continuous linear map <math>f : X \to \mathbf{K}</math> such that <math>f(z) = \|z\|</math> and <math>\|f\| \leq 1.</math>  This implies that the [[Reflexive space#Normed spaces|natural injection]] <math>J</math> from a normed space {{mvar|X}} into its [[Reflexive space#Normed spaces|double dual]] <math>V^{**}</math> is isometric.

That last result also suggests that the Hahn–Banach theorem can often be used to locate a "nicer" topology in which to work.  For example, many results in functional analysis assume that a space is [[Hausdorff space|Hausdorff]] or [[Locally convex space|locally convex]].  However, suppose {{mvar|X}} is a topological vector space, not necessarily Hausdorff or [[Locally convex topological vector space|locally convex]], but with a nonempty, proper, convex, open set {{mvar|M}}.  Then geometric Hahn–Banach implies that there is a hyperplane separating {{mvar|M}} from any other point.  In particular, there must exist a nonzero functional on {{mvar|X}} — that is, the [[Dual space#Continuous dual space|continuous dual space]] <math>X^*</math> is non-trivial.{{sfn|Narici|Beckenstein|2011|pp=177-220}}{{sfn|Schaefer|Wolff|1999|p=47}} Considering {{mvar|X}} with the [[weak topology]] induced by <math>X^*,</math> then {{mvar|X}} becomes locally convex; by the second bullet of geometric Hahn–Banach, the weak topology on this new space [[separates points]]. 
Thus {{mvar|X}} with this weak topology becomes [[Hausdorff space|Hausdorff]].  This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

===Partial differential equations===

The Hahn–Banach theorem is often useful when one wishes to apply the method of [[a priori estimate]]s.  Suppose that we wish to solve the linear differential equation <math>P u = f</math> for <math>u,</math> with <math>f</math> given in some Banach space {{mvar|X}}.  If we have control on the size of <math>u</math> in terms of <math>\|f\|_X</math> and we can think of <math>u</math> as a bounded linear functional on some suitable space of test functions <math>g,</math> then we can view <math>f</math> as a linear functional by adjunction: <math>(f, g) = (u, P^*g).</math>  At first, this functional is only defined on the image of <math>P,</math> but using the Hahn–Banach theorem, we can try to extend it to the entire codomain {{mvar|X}}.  The resulting functional is often defined to be a [[Weak solution|weak solution to the equation]].

===Characterizing reflexive Banach spaces===

{{Math theorem
| name = Theorem{{sfn|Narici|Beckenstein|2011|p=212}}
| math_statement = A real Banach space is [[Reflexive space|reflexive]] if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.
}}

===Example from Fredholm theory===
To illustrate an actual application of the Hahn–Banach theorem, we will now prove a result that follows almost entirely from the Hahn–Banach theorem.

{{Math theorem
| name = Proposition
| math_statement = Suppose <math>X</math> is a Hausdorff locally convex TVS over the field <math>\mathbf{K}</math> and <math>Y</math> is a vector subspace of <math>X</math> that is [[TVS isomorphism|TVS–isomorphic]] to <math>\mathbf{K}^I</math> for some set <math>I.</math> 
Then <math>Y</math> is a closed and [[Complemented subspace|complemented]] vector subspace of <math>X.</math>
}}

{{Math proof|drop=hidden|proof=
Since <math>\mathbf{K}^I</math> is a complete TVS so is <math>Y,</math> and since any complete subset of a Hausdorff TVS is closed, <math>Y</math> is a closed subset of <math>X.</math> 
Let <math>f = \left(f_i\right)_{i \in I} : Y \to \mathbf{K}^I</math> be a TVS isomorphism, so that each <math>f_i : Y \to \mathbf{K}</math> is a continuous surjective linear functional. 
By the Hahn–Banach theorem, we may extend each <math>f_i</math> to a continuous linear functional <math>F_i : X \to \mathbf{K}</math> on <math>X.</math> 
Let <math>F := \left(F_i\right)_{i \in I} : X \to \mathbf{K}^I</math> so <math>F</math> is a continuous linear surjection such that its restriction to <math>Y</math> is <math>F\big\vert_Y = \left(F_i\big\vert_Y\right)_{i \in I} = \left(f_i\right)_{i \in I} = f.</math> 
Let <math>P := f^{-1} \circ F : X \to Y,</math> which is a continuous linear map whose restriction to <math>Y</math> is <math>P\big\vert_Y = f^{-1} \circ F\big\vert_Y = f^{-1} \circ f = \mathbf{1}_Y,</math> where <math>\mathbb{1}_Y</math> denotes the [[identity map]] on <math>Y.</math> 
This shows that <math>P</math> is a continuous [[linear projection]] onto <math>Y</math> (that is, <math>P \circ P = P</math>). 
Thus <math>Y</math> is complemented in <math>X</math> and <math>X = Y \oplus \ker P</math> in the category of TVSs. <math>\blacksquare</math>
}}

The above result may be used to show that every closed vector subspace of <math>\R^{\N}</math> is complemented because any such space is either finite dimensional or else TVS–isomorphic to <math>\R^{\N}.</math>