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:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“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)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)