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:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)