The Sandwich Theorem for Sublinear and Superlinear Functionals

TL;DR

This paper generalizes the Hahn-Banach theorem by establishing conditions under which a linear functional can be extended to the whole space while being bounded above by a sublinear functional and below by a superlinear functional.

Contribution

It introduces a new extension theorem for linear functionals dominated by sublinear and superlinear functionals simultaneously, broadening the classical Hahn-Banach theorem.

Findings

Extension of linear functionals under combined sublinear and superlinear bounds

Generalization of Hahn-Banach theorem to more complex domination conditions

Theoretical framework for duality in vector spaces with mixed bounds

Abstract

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.