). Relation to linear functionals If is a sublinear function on a real vector space then the following are equivalent: is a linear functional. for every for every is a minimal sublinear function. If is a sublinear function on a real vector space then there exists a linear functional on such that If is a real vector space, is a linear functional on and is a positive sublinear function on then on if and only if Dominating a linear functional A real-valued function defined on a subset of a real or complex vector space is said to be a sublinear function if for every that belongs to the domain of If is a real linear functional on then is dominated by (that is, ) if and only if Moreover, if is a seminorm or some other (which by definition means that holds for all ) then if and only if Continuity Suppose is a topological vector space (TVS) over the real or complex numbers and is a sublinear function on Then the following are equivalent: is continuous; is continuous at 0; is uniformly continuous on ; and if is positive then this list may be extended to include: is open in If is a real TVS, is a linear functional on and is a continuous sublinear function on then on implies that is continuous. Relation to Minkowski functions and open convex sets Relation to open convex sets Operators 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. Computer science definition In computer science, a function is called sublinear if or in asymptotic notation (notice the small ). Formally, if and only if, for any given there exists an such that for That is, grows slower than any linear function. The two meanings should not be confused: while a Banach functional is convex, almost the opposite is true for functions of sublinear growth: every function can be upper-bounded by a concave function of sublinear growth. See also Notes References Bibliography Articles containing proofs Functional analysis Linear algebra Types of functions