What is "unified convergence?"?

Question

I have seen it used in the scientific literature but can't find a simple definition of just this specific term.

Answer 1

You probably mean "uniform convergence". The idea is that in regular, non-uniform convergence of a sequence of functions at each point x one can guarantee that fn(x) is close to f(x) simply by n being large enough, say bigger than N. In general N depends on x, but in uniform convergence N does not depend on x.

Example: Let fn(x)=(x+1/n)^2. Then fn(x) converges to x^2 but not uniformly since the bigger x is the bigger n has to be to guarantee that fn(x) is a given closeness to x^2 since (x+1/n)^2 - x^2 depends on x and n.

Example: Let fn(x) = x+1/n. Then fn(x) converges uniformly to x since (x+1/n) - x does not depend on x.