  The Hahn-Banach theorem is an extension theorem for linear functionals which
preserves certain properties. Specifically, if a linear functional is defined
on a subspace of a real vector space which is dominated by a sublinear
functional on the entire space, then this functional can be extended to a
linear functional on the entire space which is still dominated by a sublinear
functional. In this paper, we generalize this result to show that a linear
functional defined on a subspace of a real vector space which is dominated by a
sublinear functional and also dominates a superlinear functional on the entire
space can be extended to a linear functional on the entire space which is also
dominated by a sublinear functional and dominates a superlinear functional.
