Definition:Uniform Convergence

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Definition

Metric Space

Let $S$ be a set.

Let $M = \struct {A, d}$ be a metric space.

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

Let:

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

Then $\sequence {f_n}$ 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:

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

That is:

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

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

Infinite Series

Let $S \subseteq \R$.

Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.

Let $\sequence {s_n}$ be sequence of real functions $S \to \R$ with:

$\ds \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_n} x$

for each $n \in \N$ and $x \in S$.

We say that:

$\ds \sum_{n \mathop = 1}^\infty f_n$

converges uniformly to a real function $f: S \to \R$ on $S$ if $\sequence {s_n}$ converges uniformly to $f$ on $S$.

Also defined as

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

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.