Submultiplicativity

Submultiplicativity is a property of a function or norm in mathematics. Formally, a function f defined on a multiplicative semigroup S is submultiplicative if f(xy) ≤ f(x) f(y) for all x, y in S. In the setting of normed algebras, a norm ||·|| is submultiplicative if ||xy|| ≤ ||x|| ||y|| for all x, y in the algebra.

Examples include the absolute value on real numbers, which satisfies |ab| ≤ |a||b|, and the operator norm on matrices, for which ||AB|| ≤ ||A|| ||B|| holds for compatible matrices. Submultiplicativity is a key condition for norms used to study products and powers of elements in an algebra.

Submultiplicative sequences are another important instance. A sequence (a_n) with a_0 = 1 is submultiplicative if a_{m+n} ≤ a_m a_n for all m, n. A fundamental result, Fekete’s lemma, implies that the limit lim_{n→∞} a_n^{1/n} exists and equals inf_n a_n^{1/n}. From the inequality, one derives a_n ≤ a_1^n (by induction) when a_0 = 1, illustrating controlled exponential growth.

Submultiplicativity often contrasts with subadditivity and with strict multiplicativity (where f(xy) = f(x) f(y)). It underpins stability and growth estimates in analysis, linear algebra, and combinatorics, and it is central to the theory of Banach algebras and related areas.