Let \(X\) be topological space and \(Y\) a metric space . A sequence of functions \(f_n\colon X\to Y\) \(X\) converges locally uniformly with limit \(f\colon X\to Y\) if for every \(z\in X\) a neighbourhood \(V\subset X\) exists, such that \((f_n)\) converges uniformly on \(V\cap X\) with limit \(f|_{V\cap X}\).
Remarks
- Uniform convergence implies local uniform convergence.
- Local uniform convergence implies pointwise convergence .
- Limits of local uniform convergent sequences of continuous functions are continuous.
- In some literature local uniform convergence means uniform convergence on compact subsets.