Definition
Suppose S is a set and fn : S → R is a real-valued function for every natural number n. We say that the sequence (fn)n∈N is uniformly convergent with limit f : S → R if for every ε > 0, there exists a natural number N such that for all x ∈ S and all n ≥ N we have |fn(x) − f(x)| < ε.
Consider the sequence αn = supx |fn(x) − f(x)| where the supremum is taken over all x ∈ S. Clearly fn converges to f uniformly if and only if αn tends to 0.
The sequence (fn)n∈N 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
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)