positive nor a seminorm; the same is true of this map's negation More generally, for any real the map is a sublinear function on and moreover, every sublinear function is of this form; specifically, if and then and If and are sublinear functions on a real vector space then so is the map More generally, if is any non-empty collection of sublinear functionals on a real vector space and if for all then is a sublinear functional on A function is sublinear if and only if it is subadditive, convex, and satisfies Properties Every sublinear function is a convex function: For If is a sublinear function on a vector space then for every which implies that at least one of and must be nonnegative; that is, for every Moreover, when is a sublinear function on a real vector space then the map defined by is a seminorm. Subadditivity of guarantees that for all vectors so if is also symmetric then the reverse triangle inequality will hold for all vectors Defining then subadditivity also guarantees that for all the value of on the set is constant and equal to In particular, if is a vector subspace of then and the assignment which will be denoted by is a well-defined real-valued sublinear function on the quotient space that satisfies If is a seminorm then is just the usual canonical norm on the quotient space Adding to both sides of the hypothesis (where ) and combining that with the conclusion gives which yields many more inequalities, including, for instance, in which an expression on one side of a strict inequality can be obtained from the other by replacing the symbol with (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged). Associated seminorm If is a real-valued sublinear function on a real vector space (or if is complex, then when it is considered as a real vector space) then the map defines a seminorm on the real vector space called the seminorm associated with A sublinear function on a real or complex vector space is a symmetric function if and only if where as before. More generally, if is a real-valued sublinear function on a (real or complex) vector space then will define a seminorm on if this supremum is always a real number (that is, never equal to