Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous.
Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm except that it is not required to map non-zero vectors to non-zero values.
The notion of a sublinear function was introduced by Stefan Banach when he proved his version of the Hahn-Banach theorem.
[1] There is also a different notion in computer science, described below, that also goes by the name "sublinear function."
Every subadditive symmetric function is necessarily nonnegative.
[proof 1] A sublinear function on a real vector space is symmetric if and only if it is a seminorm.
are sublinear functions on a real vector space
is any non-empty collection of sublinear functionals on a real vector space
, any two properties among subadditivity, convexity, and positive homogeneity implies the third.
is a sublinear function on a real vector space then the map
is also symmetric then the reverse triangle inequality will hold for all vectors
is a well-defined real-valued sublinear function on the quotient space
is just the usual canonical norm on the quotient space
(or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).
is a real-valued sublinear function on a real vector space
is complex, then when it is considered as a real vector space) then the map
defines a seminorm on the real vector space
on a real or complex vector space is a symmetric function if and only if
is a real-valued sublinear function on a (real or complex) vector space
is a sublinear function on a real vector space
is a sublinear function on a real vector space
defined on a subset of a real or complex vector space
is a seminorm or some other symmetric map (which by definition means that
be a sublinear function on a real vector space
is a topological vector space (TVS) over the real or complex numbers and
is a convex open neighborhood of the origin in a topological vector space
is a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers.
The concept can be extended to operators that are homogeneous and subadditive.
This requires only that the codomain be, say, an ordered vector space to make sense of the conditions.
can be upper-bounded by a concave function of sublinear growth.