The Hahn–Banach–Lagrange theorem

The idea behind this article is to provide a unified and relatively nontechnical framework for treating the main existence theorems for continuous linear functionals in functional analysis, convex analysis, Lagrange multiplier theory, and minimax theory. While many of the results in this article are already known, our approach is new, and gives a large number of results with considerable economy of effort. More specifically, we prove a version of the Hahn–Banach theorem, the ‘‘Hahn–Banach–Lagrange theorem’’, that is sufficiently strong that all the aforementioned results follow fairly directly from it. Many of the results in this article, which are already known, are usually proved using the Eidelheit separation theorem in a product space. The proofs of these results given here avoid the problem of the elimination of the ‘‘vertical hyperplane’’. Furthermore, our approach leads naturally to the sharp numerical bounds for various problems discussed in sections 6, 8, and 10. (The sharp numerical bound discussed in