An approximate Hahn–Banach theorem for positively homogeneous functions

Abstract

This note provides an approximate version of the Hahn–Banach theorem for non-necessarily convex extended-real valued positively homogeneous functions of degree one. Given such a function defined on the real vector space , and a linear function defined on a subspace of and dominated by (i.e. for all ), we say that can approximately be -extended to , if is the pointwise limit of a net of linear functions on , every one of which can be extended to a linear function defined on and dominated by . The main result of this note proves that can approximately be -extended to if and only if is dominated by , the pointwise supremum over the family of all the linear functions on which are dominated by .