Uniform Convergence - Definition

Definition

Suppose S is a set and fn : SR is a real-valued function for every natural number n. We say that the sequence (fn)nN is uniformly convergent with limit f : SR if for every ε > 0, there exists a natural number N such that for all xS and all nN we have |fn(x) − f(x)| < ε.

Consider the sequence αn = supx |fn(x) − f(x)| where the supremum is taken over all xS. Clearly fn converges to f uniformly if and only if αn tends to 0.

The sequence (fn)nN is said to be locally uniformly convergent with limit f if for every x in some metric space S, there exists an r > 0 such that (fn) converges uniformly on B(x,r) ∩ S.

Read more about this topic:  Uniform Convergence

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