Definition:Uniform Convergence

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Definition

Metric Space

Let $S$ be a set.

Let $M = \left({A, d}\right)$ be a metric space.

Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: S \to A$.

Let:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in S: d \left({f_n \left({x}\right), f \left({x}\right)}\right) < \epsilon$


Then $\left \langle {f_n} \right \rangle$ converges to $f$ uniformly on $S$ as $n \to \infty$.


Real Numbers

The above definition can be applied directly to the real numbers treated as a metric space:


Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$.

Let:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in D: \size {\map {f_n} x - \map f x} < \epsilon$


Then $\sequence {f_n}$ converges to $f$ uniformly on $D$ as $n \to \infty$.


Also defined as

Some sources insist that $N \in \N$ but this is unnecessary and makes proofs more cumbersome.


Also see


Comment

Note that this definition of convergence of a function is stronger than that for pointwise convergence, in which it is necessary to specify a value of $N$ given $\epsilon$ for each individual point.

In uniform convergence, given $\epsilon$ you need to specify a value of $N$ which holds for all points in the domain of the function.


Historical Note

The concept of uniform convergence was created by Karl Weierstrass during his investigation of power series.