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 Sequences
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 and only if $\sequence {s_n}$ converges uniformly to $f$ on $S$.
Also defined as
In definining uniform convergence, some sources insist that $N \in \N$, but this is unnecessary and makes proofs more cumbersome.
Also see
- Results about uniform convergence can be found here.
Comment
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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.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): uniform convergence
- 2000: James R. Munkres: Topology (2nd ed.): $\S 21$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): uniform convergence
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): convergence of functions
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): uniform convergence