Big O Notation - Generalizations and Related Usages

Generalizations and Related Usages

The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any topological group is also possible. The "limiting process" x→x_o can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g. The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,

which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to if f and g are positive real valued functions.) For example, 2x is Θ(x), but 2x − x is not o(x).