Editing Hahn–Banach theorem (section)

===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]].