Let \((f_n)\) be a sequence of functions and \(Y\) a metric space . We say the sequence is uniformly convergent if a function \(f\colon X\to Y\) exists, such that for every \(\varepsilon>0\) exists a \(n_0>0\) with
\begin{equation*} \sup_{x\in X} d(f_n(x),f(x))< \varepsilon \end{equation*}for all \(n\ge n_0\).
Notation: \(f_n\xrightarrow{\text{unif.}}f\)
Remarks
- Uniform convergence implies pointwise convergence .
- Uniform convergence implies local uniform convergence .
- Uniform convergence preserves certain regularity properties of functions in the sequence.