The Hahn–Banach theorem is often useful when one wishes to apply the method of a priori estimates. Suppose that we wish to solve the linear differential equation for with given in some Banach space . If we have control on the size of in terms of and we can think of as a bounded linear functional on some suitable space of test functions then we can view as a linear functional by adjunction: At first, this functional is only defined on the image of but using the Hahn–Banach theorem, we can try to extend it to the entire codomain . The resulting functional is often defined to be a weak solution to the equation.
The Hahn–Banach theorem is often useful when one wishes to apply the method of a priori estimates. Suppose that we wish to solve the li,near differential equation for ?with given in some Banach space . If we have control on the size of in terms of and we can think of as a bounded linear functional on some suitable space of test functions then we can view as a linear functional by adjunction: At first, this functional is only defined on the image of but using, the Hahn–Banach theorem, we can try to extend it to the entire codomain . The resulting functional is often defined to be a weak solution to the equation.