# Definition: Pointwise and Uniform Convergence

Let $$(X,d)$$ and $$(Y,d)$$ be metric spaces, let $$f:X\to Y$$ be a function and let $$f_n:X\to Y$$, $$n\in\mathbb N$$ be a sequence of functions.

• The sequence $$(f_n)_{n\in\mathbb N}$$ converges pointwise to $$f$$, if for each $$x\in X$$ and each $$\epsilon > 0$$ there exists an $$N(x,\epsilon)\in\mathbb N$$, such that $d(f_n(x),f(x)) < \epsilon \text{ for all } n\ge N(x,\epsilon),$ i.e. if the sequence $$(f_n(x))_{n\in\mathbb N}$$ is convergent to $$f(x)$$ for each $$x\in X$$.

• The sequence $$(f_n)_{n\in\mathbb N}$$ converges uniformly to $$f$$, if for each $$\epsilon > 0$$ there exists an $$N(\epsilon)\in\mathbb N$$, such that $d(f_n(x),f(x)) < \epsilon \text{ for all } x\in X\text{ and all } n\ge N(\epsilon),$ i.e. if the sequence $$(f_n(x))_{n\in\mathbb N}$$ is convergent to $$f(x)$$ for each $$x\in X$$ in such a way that the distance from $$f_n(x)$$ to $$f(x)$$ for sufficient big index $$n$$ does not exceed the threshold value $$\epsilon$$ for all $$x\in X$$.

Please note that the difference between pointwise and uniform convergence is that in the first case the index $$N$$ depends on both, $$x$$ and $$\epsilon$$, while in the latter case it only depends on $$\epsilon$$.

Examples: 1 Corollaries: 1

Corollaries: 1
Examples: 2

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984