In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If is a subset of a real or complex vector space then the Minkowski functional or gauge of is defined to be the function valued in the extended real numbers, defined by In functional analysis, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that for every this set is not empty precisely because this results in being real-valued.