In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two
elements of the
domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly
norms and
square roots.
Additive map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation:
f(x + y) = f(x) + f(y)
for every pair of elements x and y in the domain of f. For example, any linear map is additive. When ...
s are special cases of subadditive functions.
Definitions
A subadditive function is a
function , having a
domain ''A'' and an
ordered codomain ''B'' that are both
closed under addition, with the following property:
An example is the
square root function, having the
non-negative real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s as domain and codomain:
since
we have:
A
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
is called subadditive if it satisfies the
inequality
for all ''m'' and ''n''. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.
Note that while a concave sequence is subadditive, the converse is false. For example, arbitrarily assign
with values in