Let M be a nonempty, closed, subspace of a linear space X and y ∉ M. Then there exists a continuous linear functional f ∈ X* such that
f (y) > 0 and f (x) = 0 for every x ∈
As an application of the previous result, we use it in the following exercise to provide an alternative derivation of the Fredholm alternative (exercise 3.48). Note how a clever choice of space enables us to apply a separation theorem to derive an a straightforward proof of a fundamental theorem.